WIAS Preprint No. 2148, (2020)

On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection-diffusion-reaction equations


Authors

  • Frerichs, Derk
  • John, Volker
    ORCID: 0000-0002-2711-4409

2010 Mathematics Subject Classification

  • 65N30

Keywords

  • Steady-state convection-diffusion-reaction equations, convection-dominated regime, discontinuous Galerkin finite element method, reduction of spurious oscillations, post-processing approaches, slope limiters

DOI

10.20347/WIAS.PREPRINT.2769

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WIAS Preprint No. 2148, (2020)

Multiscale thermodynamics of charged mixtures


Authors

  • Vágner, Petr
    ORCID: 0000-0001-5952-0025
  • Pavelka, Michal
  • Esen, Oğul
    ORCID: 0000-0002-6766-0287

2010 Mathematics Subject Classification

  • 78A25 35Q61 82B35

Keywords

  • GENERIC, electrodynamics, continuum mechanics, non-equilibrium thermodynamics, polarization, magnetization, multiscale modeling, hamiltonian mechanics

DOI

10.20347/WIAS.PREPRINT.2733

Abstract

A multiscale theory of interacting continuum mechanics and thermodynamics of mixtures of fluids, electrodynamics, polarization and magnetization is proposed. The mechanical (reversible) part of the theory is constructed in a purely geometric way by means of semidirect products. This leads to a complex Hamiltonian system with a new Poisson bracket, which can be used in principle with any energy functional. The thermodynamic (irreversible) part is added as gradient dynamics, generated by derivatives of a dissipation potential, which makes the theory part of the GENERIC framework. Subsequently, Dynamic MaxEnt reductions are carried out, which lead to reduced GENERIC models for smaller sets of state variables. Eventually, standard engineering models are recovered as the low-level limits of the detailed theory. The theory is then compared to recent literature.

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WIAS Preprint No. 2148, (2020)

An assessment of two classes of variational multiscale methods for the simulation of incompressible turbulent flows


Authors

  • Ahmed, Naveed
    ORCID: 0000-0002-9322-0373
  • John, Volker
    ORCID: 0000-0002-2711-4409

2010 Mathematics Subject Classification

  • 76F65

Keywords

  • incompressible turbulent flows, residual-based VMS method, SUPG method, projection-based VMS method, turbulent channel flows, LSC preconditioner

DOI

10.20347/WIAS.PREPRINT.2698

Abstract

A numerical assessment of two classes of variational multiscale (VMS) methods for the simulation of incompressible flows is presented. Two types of residual-based VMS methods and two types of projection-based VMS methods are included in this assessment. The numerical simulations are performed at turbulent channel flow problems with various friction Reynolds numbers. It turns out the the residual-based VMS methods, in particular when used with a pair of inf-sup stable finite elements, give usually the most accurate results for second order statistics. For this pair of finite element spaces, a flexible GMRES method with a Least Squares Commutator (LSC) preconditioner proved to be an efficient solver.

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WIAS Preprint No. 2148, (2020)

A theory of generalised solutions for ideal gas mixtures with Maxwell--Stefan diffusion


Authors

  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500

2010 Mathematics Subject Classification

  • 35M33 35Q30 76N10 35D30 35Q35 35Q79 76R50

Keywords

  • Multicomponent flow, ideal mixture, compressible fluid, diffusion, system of mixed-type, global-in-time existence, weak solutions, transport coefficients

DOI

10.20347/WIAS.PREPRINT.2700

Abstract

After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic-hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids emphex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

Appeared in

  • Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), pp. 4035--4067(published online in Nov 2020), DOI 10.3934/dcdss.2020458 .

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WIAS Preprint No. 2148, (2020)

On accelerated alternating minimization


Authors

  • Guminov, Sergey
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Gasnikov, Alexander

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25

Keywords

  • Convex optimization, acceleration, non-convex optimization, alternating minimization

DOI

10.20347/WIAS.PREPRINT.2695

Abstract

Alternating minimization (AM) optimization algorithms have been known for a long time and are of importance in machine learning problems, among which we are mostly motivated by approximating optimal transport distances. AM algorithms assume that the decision variable is divided into several blocks and minimization in each block can be done explicitly or cheaply with high accuracy. The ubiquitous Sinkhorn's algorithm can be seen as an alternating minimization algorithm for the dual to the entropy-regularized optimal transport problem. We introduce an accelerated alternating minimization method with a $1/k^2$ convergence rate, where $k$ is the iteration counter. This improves over known bound $1/k$ for general AM methods and for the Sinkhorn's algorithm. Moreover, our algorithm converges faster than gradient-type methods in practice as it is free of the choice of the step-size and is adaptive to the local smoothness of the problem. We show that the proposed method is primal-dual, meaning that if we apply it to a dual problem, we can reconstruct the solution of the primal problem with the same convergence rate. We apply our method to the entropy regularized optimal transport problem and show experimentally, that it outperforms Sinkhorn's algorithm.

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WIAS Preprint No. 2148, (2020)

Asymptotic study of the electric double layer at the interface of a polyelectrolyte gel and solvent bath


Authors

  • Hennessy, Matthew G.
  • Celora, Giulia L.
  • Münch, Andreas
  • Waters, Sarah L.
  • Wagner, Barbara
    ORCID: 0000-0001-8306-3645

2010 Mathematics Subject Classification

  • 74A30 80A22 34B15

2010 Physics and Astronomy Classification Scheme

  • 83.80.Rs; 83.10.Tv

Keywords

  • Polyelectrolyte gel, phase separation, matched asymptotic expansions

DOI

10.20347/WIAS.PREPRINT.2751

Abstract

An asymptotic framework is developed to study electric double layers that form at the inter-face between a solvent bath and a polyelectrolyte gel that can undergo phase separation. The kinetic model for the gel accounts for the finite strain of polyelectrolyte chains, free energy ofinternal interfaces, and Stefan?Maxwell diffusion. By assuming that the thickness of the doublelayer is small compared to the typical size of the gel, matched asymptotic expansions are used toderive electroneutral models with consistent jump conditions across the gel-bath interface in two-dimensional plane-strain as well as fully three-dimensional settings. The asymptotic frameworkis then applied to cylindrical gels that undergo volume phase transitions. The analysis indicatesthat Maxwell stresses are responsible for generating large compressive hoop stresses in the double layer of the gel when it is in the collapsed state, potentially leading to localised mechanicalinstabilities that cannot occur when the gel is in the swollen state. When the energy cost of in-ternal interfaces is sufficiently weak, a sharp transition between electrically neutral and chargedregions of the gel can occur. This transition truncates the double layer and causes it to have finitethickness. Moreover, phase separation within the double layer can occur. Both of these featuresare suppressed if the energy cost of internal interfaces is sufficiently high. Thus, interfacial freeenergy plays a critical role in controlling the structure of the double layer in the gel.

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WIAS Preprint No. 2148, (2020)

Dimension reduction of thermistor models for large-area organic light-emitting diodes


Authors

  • Glitzky, Annegret
  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Nika, Grigor
    ORCID: 0000-0002-4403-6908

2010 Mathematics Subject Classification

  • 35J92 35Q79 35J57 80A20 35B30 35B20

Keywords

  • Thermistor system, dimension reduction, Joule heat, organic light-emitting diode, thin-film devices, multi-scale limit

DOI

10.20347/WIAS.PREPRINT.2719

Abstract

An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional φ(x)-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter ε > 0 which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.

Appeared in

  • Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), pp. 3953--3971 (published online on 28.11.2020), DOI 10.3934/dcdss.2020460 .

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WIAS Preprint No. 2148, (2020)

A dynamical theory for singular stochastic delay differential equations II: Nonlinear equations and invariant manifolds


Authors

  • Ghani Varzaneh, Mazyar
  • Riedel, Sebastian

2010 Mathematics Subject Classification

  • 34K19 34K50 37D10 37H15 60H20

Keywords

  • Random dynamical systems, rough paths, stable and unstable manifolds, stochastic delay differential equations

DOI

10.20347/WIAS.PREPRINT.2701

Abstract

Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [GVR].

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WIAS Preprint No. 2148, (2015)

On microscopic origins of generalized gradient structures


Authors

  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Peletier, Mark A.
    ORCID: 0000-0001-9663-3694
  • Renger, D. R. Michiel
    ORCID: 0000-0003-3557-3485

2010 Mathematics Subject Classification

  • 35K55 35Q82 49S05 49J40 49J45 60F10 60J25

Keywords

  • Generalized gradient structure, gradient system, evolutionary Gamma-convergence, energy-dissipation principle, variational evolution, relative entropy, large-deviation principle

DOI

10.20347/WIAS.PREPRINT.2148

Abstract

Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

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WIAS Preprint No. 2148, (2020)

Adaptive manifold clustering


Authors

  • Besold, Franz
  • Spokoiny, Vladimir
    ORCID: 0000-0002-2040-3427

2010 Mathematics Subject Classification

  • 62H30 62G10

Keywords

  • Adaptive weights, likelihood-ratio test, nonparametric clustering, manifold, reach

Abstract

Clustering methods seek to partition data such that elements are more similar to elements in the same cluster than to elements in different clusters. The main challenge in this task is the lack of a unified definition of a cluster, especially for high dimensional data. Different methods and approaches have been proposed to address this problem. This paper continues the study originated by [6] where a novel approach to adaptive nonparametric clustering called Adaptive Weights Clustering (AWC) was offered. The method allows analyzing high-dimensional data with an unknown number of unbalanced clusters of arbitrary shape under very weak modeling as-sumptions. The procedure demonstrates a state-of-the-art performance and is very efficient even for large data dimension D. However, the theoretical study in [6] is very limited and did not re-ally address the question of efficiency. This paper makes a significant step in understanding the remarkable performance of the AWC procedure, particularly in high dimension. The approach is based on combining the ideas of adaptive clustering and manifold learning. The manifold hypoth-esis means that high dimensional data can be well approximated by a d-dimensional manifold for small d helping to overcome the curse of dimensionality problem and to get sharp bounds on the cluster separation which only depend on the intrinsic dimension d. We also address the problem of parameter tuning. Our general theoretical results are illustrated by some numerical experiments.

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WIAS Preprint No. 2148, (2020)

On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion


Authors

  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Ernst, Oliver
  • Sprungk, Björn
  • Tamellini, Lorenzo

2010 Mathematics Subject Classification

  • 65D05 65D15 65C30 60H25

Keywords

  • Random PDEs, parametric PDEs, a posteriori adaptivity, residual error estimator, convergence, affine diffusion coefficient, sparse grids, stochastic collocation

DOI

10.20347/WIAS.PREPRINT.2753

Abstract

Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting.

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WIAS Preprint No. 2148, (2020)

Fast Raman mapping and in situ TEM observation of metal induced crystallization of amorphous silicon


Authors

  • Uebel, David
  • Kayser, Stefan
  • Markurt, Toni
  • Ernst, Owen C.
  • Teubner, Thomas
  • Boeck, Torsten

2010 Physics and Astronomy Classification Scheme

  • 81.10.-h

Keywords

  • Crystal growth, Raman mapping, in situ transmission electron microspcopy, metal-induced crys-tallization, silicon for photovoltaics

DOI

10.20347/WIAS.PREPRINT.2772

Abstract

Crystalline silicon is grown onto an amorphous silicon (a-Si) seed layer from liquid tin solution (steady state liquid phase epitaxy, SSLPE). To investigate the crystallization of embedded a-Si during our process, we adapted Raman measurements for fast mapping, with dwell times of just one second per single measurement. A purposely developed imaging algorithm which performs point-by-point gauss fitting provides adequate visualization of the data. We produced scans of a-Si layers showing crystalline structures formed in the a-Si matrix during processing. Compared to scanning electron microscopy images which reveal merely the topography of the grown layer, new insights are gained into the role of the seed layer by Raman mapping. As part of a series of SSLPE experiments, which were interrupted at various stages of growth, we show that plate-like crystallites grow laterally over the a-Si layer while smaller, randomly orientated crystals arise from the a-Si layer. Results are confirmed by an in situ TEM experiment of the metal-induced crystallization. Contrary to presumptions, initially formed surface crystallites do not originate from the seed layer and are irrelevant to the final growth morphology, since they dissolve within minutes due to Ostwald ripening. The a-Si layer crystallizes within minutes as well, and crystallites of the final morphology originate from seeds of this layer.

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WIAS Preprint No. 2148, (2020)

Approximation schemes for materials with discontinuities


Authors

  • Bartels, Sören
  • Milicevic, Marijo
  • Thomas, Marita
    ORCID: 0000-0001-9172-014X
  • Tornquist, Sven
  • Weber, Nico

2010 Mathematics Subject Classification

  • 35K86 74R05 49J45 49S05 65M12 65M60 74H10 74H20 74H30 35M86 35Q74

Keywords

  • Visco-elastodynamic damage, Ambrosio-Tortorelli model for phase-field fracture, viscous evolution, rate-independent limit, partial damage, damage evolution with gradient regularization, semistable energetic solutions, numerical approximation, iterative solution, damage evolution with spatial regularization, functions of bounded variation

DOI

10.20347/WIAS.PREPRINT.2799

Abstract

Damage and fracture phenomena are related to the evolution of discontinuities both in space and in time. This contribution deals with methods from mathematical and numerical analysis to handle these: Suitable mathematical formulations and time-discrete schemes for problems with discontinuities in time are presented. For the treatment of problems with discontinuities in space, the focus lies on FE-methods for minimization problems in the space of functions of bounded variation. The developed methods are used to introduce fully discrete schemes for a rate-independent damage model and for the viscous approximation of a model for dynamic phase-field fracture. Convergence of the schemes is discussed.

Appeared in

  • Non-standard Discretisation Methods in Solid Mechanics, J. Schröder, P. Wriggers, eds., vol. 98 of Lecture Notes in Applied and Computational Mechanics, Springer, Cham, 2022, pp. 505--565, DOI 10.1007/978-3-030-92672-4_17 .

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WIAS Preprint No. 2148, (2020)

Revealing all states of dewetting of a thin gold layer on a silicon surface by nanosecond laser conditioning


Authors

  • Ernst, Owen C.
  • Uebel, David
  • Kayser, Stefan
  • Lange, Felix
  • Teubner, Thomas
  • Boeck, Torsten

Keywords

  • laser matter interaction, dewetting, metal nanoparticles, Mikowski measure, heat simulation thin metal layers

DOI

10.20347/WIAS.PREPRINT.2777

Abstract

Dewetting is a ubiquitous phenomenon which can be applied to the laser synthesis of nanoparticles. A classical spinodal dewetting process takes place in four successive states, which differ from each other in their morphology. In this study all states are revealed by interaction of pulsed nanosecond UV laser light with thin gold layers with thicknesses between 1 nm and 10 nm on (100) silicon wafers. The specific morphologies of the dewetting states are discussed with particular emphasis on the state boundaries. The main parameter determining which state is formed is not the duration for which the gold remains liquid, but rather the input energy provided by the laser. This shows that each state transition has a separate measurable activation energy. The temperature during the nanosecond pulses and the duration during which the gold remains liquid was determined by simulation using the COMSOL Multiphysics software package. Using these calculations, an accurate local temperature profile and its development over time was simulated. An analytical study of the morphologies and formed structures was performed using Minkowski measures. With aid of this tool, the laser induced structures were compared with thermally annealed samples, with perfectly ordered structures and with perfectly random structures. The results show that both, structures of the laser induced and the annealed samples, strongly resemble the perfectly ordered structures. This reveals a close relationship between these structures and suggests that the phenomenon under investigation is indeed a spinodal dewetting generated by an internal material wave function.
The purposeful generation of these structures and the elucidation of the underlying mechanism of dewetting by ultrashort pulse lasers may assist the realisation of various technical elements such as nanowires in science and industry.

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WIAS Preprint No. 2148, (2020)

A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation


Authors

  • Ahmed, Naveed
    ORCID: 0000-0002-9322-0373
  • Barrenechea, Gabriel R
  • Burman, Erik
  • Guzmán, Johnny
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 65N30 65N12 76D07

Keywords

  • incompressible Navier--Stokes equations, divergence-free mixed finite element methods, pressure-robustness, convection stabilization, Galerkin least squares, vorticity equation

DOI

10.20347/WIAS.PREPRINT.2740

Abstract

Discretization of Navier--Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressureindependent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an O(hk+1/2) error estimate in the L2-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.

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WIAS Preprint No. 2148, (2020)

Tropical time series, iterated-sum signatures and quasisymmetric functions


Authors

  • Diehl, Joscha
  • Ebrahimi-Fard, Kurusch
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2010 Mathematics Subject Classification

  • 60L10 60L70 16Y60

Keywords

  • Time series analysis, time warping, tropical quasisymmetric functions, semirings

DOI

10.20347/WIAS.PREPRINT.2760

Abstract

Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects.

Appeared in

  • SIAM J. Appl. Algebr. Geom., 6:4 (2022), pp. 563--599.

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WIAS Preprint No. 2148, (2020)

Impact of the capture time on the series resistance of quantum-well diode lasers


Authors

  • Boni, Anisuzzaman
  • Wünsche, Hans-Jürgen
  • Wenzel, Hans
    ORCID: 0000-0003-1726-0223
  • Crump, Paul

2010 Mathematics Subject Classification

  • 78A60

2010 Physics and Astronomy Classification Scheme

  • 42.55.Px 42.65.Sf 73.21.Fg 73.43.Cd

Keywords

  • Semiconductor laser, capture-escape, simulation, experiment

DOI

10.20347/WIAS.PREPRINT.2735

Abstract

Electrons and holes injected into a semiconductor heterostructure containing quantum wellsare captured with a finite time. We show theoretically that this very fact can cause a considerableexcess contribution to the series resistivity and this is one of the main limiting factors to higherefficiency for GaAs based high-power lasers. The theory combines a standard microscopic-basedmodel for the capture-escape processes in the quantum well with a drift-diffusion description ofcurrent flow outside the quantum well. Simulations of five GaAs-based devices differing in theirAl-content reveal the root-cause of the unexpected and until now unexplained increase of theseries resistance with decreasing heat sink temperature measured recently. The finite capturetime results in resistances in excess of the bulk layer resistances (decreasing with increasingtemperature) from 1 mΩ up to 30 mΩ in good agreement with experiment.

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WIAS Preprint No. 2148, (2020)

Transport and continuity equations with (very) rough noise


Authors

  • Bellinger, Carlo
  • Djurdjevac, Ana
  • Friz, Peter
    ORCID: 0000-0003-2571-8388
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2010 Mathematics Subject Classification

  • 35R60 60L20 60L50

Keywords

  • Rough transport equation, rough continuitynuity equation, first order rough partial differential equations

DOI

10.20347/WIAS.PREPRINT.2696

Abstract

Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.

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WIAS Preprint No. 2148, (2020)

Modelling charge transport in perovskite solar cells: Potential-based and limiting ion depletion


Authors

  • Abdel, Dilara
    ORCID: 0000-0003-3477-7881
  • Vágner, Petr
    ORCID: 0000-0001-5952-0025
  • Fuhrmann, Jürgen
    ORCID: 0000-0003-4432-2434
  • Farrell, Patricio
    ORCID: 0000-0001-9969-6615

2010 Mathematics Subject Classification

  • 35Q81 35K57 65N08

Keywords

  • Finite volume methods, perovskite solar cells, semiconductor device modelling, drift-diffusion equations, Scharfetter--Gummel methods

DOI

10.20347/WIAS.PREPRINT.2780

Abstract

From Maxwell--Stefan diffusion and general electrostatics, we derive a drift-diffusion model for charge transport in perovskite solar cells (PSCs) where any ion in the perovskite layer may flexibly be chosen to be mobile or immobile. Unlike other models in the literature, our model is based on quasi Fermi potentials instead of densities. This allows to easily include nonlinear diffusion (based on Fermi--Dirac, Gauss--Fermi or Blakemore statistics for example) as well as limit the ion depletion (via the Fermi--Dirac integral of order-1). The latter will be motivated by a grand-canonical formalism of ideal lattice gas. Furthermore, our model allows to use different statistics for different species. We discuss the thermodynamic equilibrium, electroneutrality as well as generation/recombination. Finally, we present numerical finite volume simulations to underline the importance of limiting ion depletion.

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WIAS Preprint No. 2148, (2020)

Optimal control of a buoyancy-driven liquid steel stirring modeled with single-phase Navier--Stokes equations


Authors

  • Wilbrandt, Ulrich
  • Alia, Najib
  • John, Volker
    ORCID: 0000-0002-2711-4409

2010 Mathematics Subject Classification

  • 65M60 76F65

Keywords

  • ladle stirring, single-phase Navier--Stokes equations, turbulent incompressible flows, optimal control of PDEs, finite element method

DOI

10.20347/WIAS.PREPRINT.2776

Abstract

Gas stirring is an important process used in secondary metallurgy. It allows to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions which can be detrimental for the end-product quality. In this process, argon gas is injected from two nozzles at the bottom of the vessel and rises by buoyancy through the liquid steel thereby causing stirring, i.e., a mixing of the bath. The gas flow rates and the positions of the nozzles are two important control parameters in practice. A continuous optimization approach is pursued to find optimal values for these control variables. The effect of the gas appears as a volume force in the single-phase incompressible NavierStokes equations. Turbulence is modeled with the Smagorinsky Large Eddy Simulation (LES) model. An objective functional based on the vorticity is used to describe the mixing in the liquid bath. Optimized configurations are compared with a default one whose design is based on a setup from industrial practice.

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WIAS Preprint No. 2148, (2020)

Data-driven confidence bands for distributed nonparametric regression


Authors

  • Avanesov, Valeriy

2010 Mathematics Subject Classification

  • 62G15 62F40 60G15

Keywords

  • Gaussian process regression, kernel ridge regression, distributed regression, confidence bands, bootstrap

DOI

10.20347/WIAS.PREPRINT.2729

Abstract

Gaussian Process Regression and Kernel Ridge Regression are popular nonparametric regression approaches. Unfortunately, they suffer from high computational complexity rendering them inapplicable to the modern massive datasets. To that end a number of approximations have been suggested, some of them allowing for a distributed implementation. One of them is the divide and conquer approach, splitting the data into a number of partitions, obtaining the local estimates and finally averaging them. In this paper we suggest a novel computationally efficient fully data-driven algorithm, quantifying uncertainty of this method, yielding frequentist $L_2$-confidence bands. We rigorously demonstrate validity of the algorithm. Another contribution of the paper is a minimax-optimal high-probability bound for the averaged estimator, complementing and generalizing the known risk bounds.

Appeared in

  • Proceedings of Thirty Third Conference on Learning Theory, J. Abernethy, S. Agarwal , eds., vol. 125 of Proceedings of Machine Learning Research, PMLR, 2020, pp. 300--322.

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WIAS Preprint No. 2148, (2020)

On the convexity of optimal control problems involving non-linear PDEs or VIs and applications to Nash games (changed title: Vector-valued convexity of solution operators with application to optimal control problems)


Authors

  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Stengl, Steven-Marian

2010 Mathematics Subject Classification

  • 06B99 49K21 47H04 49J40

Keywords

  • PDE-constrained optimization, K-Convexity, set-valued analysis, subdifferential, semilinear elliptic, PDEs, variational inequality

DOI

10.20347/WIAS.PREPRINT.2759

Abstract

Generalized Nash equilibrium problems in function spaces involving PDEs are considered. One of the central issues arising in this context is the question of existence, which requires the topological characterization of the set of minimizers for each player of the associated Nash game. In this paper, we propose conditions on the operator and the functional that guarantee the reduced formulation to be a convex minimization problem. Subsequently, we generalize results of convex analysis to derive optimality systems also for non-smooth operators. Our theoretical findings are illustrated by examples.

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WIAS Preprint No. 2148, (2020)

Dynamics in high-power diode lasers


Authors

  • Bandelow, Uwe
    ORCID: 0000-0003-3677-2347
  • Radziunas, Mindaugas
    ORCID: 0000-0003-0306-1266
  • Zeghuzi, Anissa
  • Wünsche, Hans-Jürgen
  • Wenzel, Hans
    ORCID: 0000-0003-1726-0223

2010 Mathematics Subject Classification

  • 78A60 35Q60 78-04 35Q79 35B35

Keywords

  • High-power diode lasers, thermal lensing, laser dynamics, heating, filaments

DOI

10.20347/WIAS.PREPRINT.2715

Abstract

High-power broad-area diode lasers (BALs) exhibit chaotic spatio-temporal dynamics above threshold. Under high power operation, where they emit tens of watts output, large amounts of heat are generated, with significant impact on the laser operation. We incorporate heating effects into a dynamical electro-optical (EO) model for the optical field and carrier dynamics along the quantum-well active zone of the laser. Thereby we effectively couple the EO and heat-transport (HT) solvers. Thermal lensing is included by a thermally-induced contribution to the index profile. The heat sources obtained with the dynamic EO-solver exhibit strong variations on short time scales, which however have only a marginal impact on the temperature distribution. We consider two limits: First, the static HT-problem, with time-averaged heat sources, which is solved iteratively together with the EO solver. Second, under short pulse operation the thermally induced index distribution can be obtained by neglecting heat flow. Although the temperature increase is small, a waveguide is introduced here within a few-ns-long pulse resulting in significant near field narrowing. We further show that a beam propagating in a waveguide structure utilized for BA lasers does not undergo filamentation due to spatial holeburning. Moreover, our results indicate that in BALs a clear optical mode structure is visible which is neither destroyed by the dynamics nor by longitudinal effects.

Appeared in

  • Proceedings of SPIE, vol. 11356, 2020, pp. 113560W-1--113560W-14, DOI 10.1117/12.2559175 .

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WIAS Preprint No. 2148, (2020)

System identification of a hysteresis-controlled pump system using SINDy


Authors

  • Thiele, Gregor
    ORCID: 0000-0002-7108-5203
  • Fey, Arne
    ORCID: 0000-0001-7351-4916
  • Sommer, David
    ORCID: 0000-0002-6797-8009
  • Krüger, Jörg
    ORCID: 0000-0001-5138-0793

2010 Mathematics Subject Classification

  • 93B30 93C10

Keywords

  • SINDyHybrid, system identification, nonlinear systems, hysteresis, hybrid dynamical systems, sparse regression

DOI

10.20347/WIAS.PREPRINT.2794

Abstract

Hysteresis-controlled devices are widely used in industrial applications. For example, cooling devices usually contain a two-point controller, resulting in a nonlinear hybrid system with two discrete states. Dynamic models of systems are essential for optimizing such industrial supply technology. However, conventional system identification approaches can hardly handle hysteresis-controlled devices. Thus, the new identification method Sparse Identification of Nonlinear Dynamics (SINDy) is extended to consider hybrid systems. SINDy composes models from basis functions out of a customized library in a data-driven manner. For modeling systems that behave dependent on their own past as in the case of natural hysteresis, Ferenc Preisach introduced the relay hysteron as an elementary mathematical description. In this new method (SINDyHybrid), tailored basis functions in form of relay hysterons are added to the library which is used by SINDy. Experiments with a hysteresis controlled water basin show that this approach correctly identifies state transitions of hybrid systems and also succeeds in modeling the dynamics of the discrete system states. A novel proximity hysteron achieves the robustness of this method. The impacts of the sampling rate and the signal noise ratio of the measurement data are examined accordingly.

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WIAS Preprint No. 2148, (2020)

How to gamble with non-stationary X-armed bandits and have no regrets


Authors

  • Avanesov, Valeriy

2010 Mathematics Subject Classification

  • 62M10 62H15

Keywords

  • Bootstrap, change point detection, nonparametrics, regression, multiscale

DOI

10.20347/WIAS.PREPRINT.2686

Abstract

In X-armed bandit problem an agent sequentially interacts with environment which yields a reward based on the vector input the agent provides. The agent's goal is to maximise the sum of these rewards across some number of time steps. The problem and its variations have been a subject of numerous studies, suggesting sub-linear and sometimes optimal strategies. The given paper introduces a new variation of the problem. We consider an environment, which can abruptly change its behaviour an unknown number of times. To that end we propose a novel strategy and prove it attains sub-linear cumulative regret. Moreover, the obtained regret bound matches the best known bound for GP-UCB for a stationary case, and approaches the minimax lower bound in case of highly smooth relation between an action and the corresponding reward. The theoretical result is supported by experimental study.

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WIAS Preprint No. 2148, (2020)

An asymptotic analysis for a generalized Cahn--Hilliard system with fractional operators


Authors

  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Gilardi, Gianni
    ORCID: 0000-0002-0651-4307
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 35K45 35K90 35R11 35B40

Keywords

  • Fractional operators, Cahn--Hilliard systems, asymptotic analysis

DOI

10.20347/WIAS.PREPRINT.2741

Abstract

In a recent paper the same authors have proved existence, uniqueness and regularity results for a class of viscous and nonviscous Cahn--Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers in the spectral sense of general linear operators, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space of square-integrable functions on a bounded and smooth three-dimensional domain, and have compact resolvents. Here, for the case of the viscous system, we analyze the asymptotic behavior of the solution as the fractional power coefficient of the second operator tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of the second operator appears.

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WIAS Preprint No. 2148, (2020)

Optimal control of a phase field system of Caginalp type with fractional operators


Authors

  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Gilardi, Gianni
    ORCID: 0000-0002-0651-4307
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 35K45 35K90 35R11 49J20 40J05 49K20

Keywords

  • Fractional operators, phase field system, nonconserved phase transition, optimal control, first-order necessary optimality conditions

DOI

10.20347/WIAS.PREPRINT.2725

Abstract

In their recent work ``Well-posedness, regularity and asymptotic analyses for a fractional phase field system'' (Asymptot. Anal. 114 (2019), 93--128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables.

Appeared in

  • Pure Appl. Funct. Anal., 7 (2022), pp. 1597--1635.

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WIAS Preprint No. 2148, (2020)

State-constrained control-affine parabolic problems II: Second order sufficient optimality conditions


Authors

  • Aronna, M. Soledad
    ORCID: 0000-0002-0640-4722
  • Bonnans, J. Frédéric
  • Kröner, Axel

2010 Mathematics Subject Classification

  • 49J20 49K20 35K58

Keywords

  • Optimal control of partial differential equations, semilinear parabolic equations, state constraints, second order analysis, Goh transform, control-affine problems

DOI

10.20347/WIAS.PREPRINT.2778

Abstract

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform.

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WIAS Preprint No. 2148, (2020)

State-constrained control-affine parabolic problems I: First and second order necessary optimality conditions


Authors

  • Aronna, M. Soledad
    ORCID: 0000-0002-0640-4722
  • Bonnans, J. Frédéric
  • Kröner, Axel

2010 Mathematics Subject Classification

  • 49J20 49K20 35J10 93C20

Keywords

  • Optimal control of partial differential equations, semilinear parabolic equations, state constraints, second order analysis, control-affine problems

DOI

10.20347/WIAS.PREPRINT.2762

Abstract

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions.

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WIAS Preprint No. 2148, (2020)

Radiation conditions for the Helmholtz equation in a half plane filled by inhomogeneous periodic material


Authors

  • Hu, Guanghui
  • Rathsfeld, Andreas
    ORCID: 0000-0002-2029-5761

2010 Mathematics Subject Classification

  • 74J20 76B15 35J50 35J08

Keywords

  • Half-space radiation condition, inhomogeneous medium, wave-mode expansion, scattering by grating, RCWA

DOI

10.20347/WIAS.PREPRINT.2726

Abstract

In this paper we consider time-harmonic acoustic wave propagation in a half-plane filled by inhomogeneous periodic medium. If the refractive index depends on the horizontal coordinate only, we define upward and downward radiating modes by solving a one-dimensional Sturm-Liouville eigenvalue problem with a complex-valued periodic coefficient. The upward and downward radiation conditions are introduced based on a generalized Rayleigh series. Using the variational method, we then prove uniqueness and existence for the scattering of an incoming wave mode by a grating located between an upper and lower half plane with such inhomogeneous periodic media. Finally, we discuss the application of the new radiation conditions to the scattering matrix algorithm, i.e., to rigorous coupled wave analysis or Fourier modal method.

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WIAS Preprint No. 2148, (2020)

Robust multiple stopping -- A path-wise duality approach


Authors

  • Laeven, Roger J. A.
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266
  • Schweizer, Nikolaus F. F.
  • Stadje, Mitja

2010 Mathematics Subject Classification

  • 49L20 60G40 91B16

Keywords

  • Optimal stopping, multiple stopping, robustness, model uncertainty, ambiguity, path-wise duality, g-expectations, BSDEs, regression

DOI

10.20347/WIAS.PREPRINT.2728

Abstract

In this paper we develop a solution method for general optimal stopping problems. Our general setting allows for multiple exercise rights, i.e., optimal multiple stopping, for a robust evaluation that accounts for model uncertainty, and for general reward processes driven by multi-dimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem which satisfy appealing path-wise optimality (almost sure) properties. Next, we exploit these theoretical results to develop upper and lower bounds which, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine upper and lower bounds. We illustrate the applicability of our general approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies.

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WIAS Preprint No. 2148, (2020)

Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations


Authors

  • Meinlschmidt, Hannes
    ORCID: 0000-0002-5874-8017
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 35J25 35B65 35R05 35Q81 92E20

Keywords

  • Elliptic regularity, nonsmooth geometry, Sneiberg stability theorem, fractional Sobolev spaces, van Roosbroeck system, semiconductor equations

DOI

10.20347/WIAS.PREPRINT.2705

Abstract

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.

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WIAS Preprint No. 2148, (2020)

Inexact model: A framework for optimization and variational inequalities


Authors

  • Stonyakin, Fedor
  • Gasnikov, Alexander
  • Tyurin, Alexander
  • Pasechnyuk, Dmitry
  • Agafonov, Artem
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Dvinskikh, Darina
  • Piskunova, Victorya

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25

2010 Physics and Astronomy Classification Scheme

  • 65K15

Keywords

  • Convex optimization, composite optimization, proximal method, level-set method, variational inequality, universal method, mirror prox, acceleration, relative smoothness

DOI

10.20347/WIAS.PREPRINT.2679

Abstract

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities.

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WIAS Preprint No. 2148, (2020)

Inexact relative smoothness and strong convexity for optimization and variational inequalities by inexact model


Authors

  • Stonyakin, Fedor
  • Gasnikov, Alexander
  • Tyurin, Alexander
  • Pasechnyuk, Dmitry
  • Agafonov, Artem
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Dvinskikh, Darina
  • Artamonov, Sergei
  • Piskunova, Victorya

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25 65K15

Keywords

  • Convex optimization, composite optimization, proximal method, level-set method, variational inequality, universal method, mirror prox, acceleration, relative smoothness, saddle-point problem

DOI

10.20347/WIAS.PREPRINT.2709

Abstract

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities.

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WIAS Preprint No. 2148, (2020)

Analysis of a compressible Stokes-flow with degenerating and singular viscosity


Authors

  • Farshbaf Shaker, Mohammad Hassan
    ORCID: 0000-0003-0543-5938
  • Thomas, Marita
    ORCID: 0000-0001-9172-014X

2010 Mathematics Subject Classification

  • 35D30 35Q35 35K92 35J70 76Nxx 76M30

Keywords

  • Suspensions, compressible flows, degenerate and singular problems

DOI

10.20347/WIAS.PREPRINT.2786

Abstract

In this paper we show the existence of a weak solution for a compressible single-phase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a density-dependent viscosity. The analysis is based on an implicit time-discrete scheme and a Galerkin-approximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of p-Laplacian type in the transport equation that allows for refined compactness arguments on subdomains.

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WIAS Preprint No. 2148, (2020)

Randomized optimal stopping algorithms and their convergence analysis


Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Belomestny, Denis
  • Hager, Paul
  • Pigato, Paolo
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266

2010 Mathematics Subject Classification

  • 60G40 65C05

Keywords

  • Randomized optimal stopping, convergence rates, Bermudan options

DOI

10.20347/WIAS.PREPRINT.2697

Abstract

In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimization algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates.

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WIAS Preprint No. 2148, (2020)

Analysis of a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model


Authors

  • Lasarzik, Robert
    ORCID: 0000-0002-1677-6925

2010 Mathematics Subject Classification

  • 35Q35 35D99 76D05

Keywords

  • Weak-strong uniqueness, phase transition, Navier--Stokes, Cahn--Hilliard, existence, thermodynamical consistent, dissipative solutions, relative energy

DOI

10.20347/WIAS.PREPRINT.2739

Abstract

In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak-strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.

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WIAS Preprint No. 2148, (2020)

Optimal control for shape memory alloys of the one-dimensional Frémond model


Authors

  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Farshbaf Shaker, Mohammad Hassan
    ORCID: 0000-0003-0543-5938
  • Shirakawa, Ken
  • Yamazaki, Noriaki

2010 Mathematics Subject Classification

  • 49J20 35K55 35R35

Keywords

  • Optimal control problem, one-dimensional Frémond model, shape memory alloys, Mosco convergence, subdifferentials

DOI

10.20347/WIAS.PREPRINT.2737

Abstract

In this paper, we consider optimal control problems for the one-dimensional Frémond model for shape memory alloys. This model is constructed in terms of basic functionals like free energy and pseudo-potential of dissipation. The state problem is expressed by a system of partial differential equations involving the balance equations for energy and momentum. We prove the existence of an optimal control that minimizes the cost functional for a nonlinear and nonsmooth state problem. Moreover, we show the necessary condition of the optimal pair by using optimal control problems for approximating systems.

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WIAS Preprint No. 2148, (2020)

Well-posedness analysis of multicomponent incompressible flow models


Authors

  • Bothe, Dieter
  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500

2010 Mathematics Subject Classification

  • 35M33 35Q30 76N10 35D35 35B65 35B35 35K57 35Q35 35Q79 76R50 80A17 80A32 92E20

Keywords

  • Multicomponent flow, complex fluid, fluid mixture, incompressible fluid, low Mach-number, strong solutions

DOI

10.20347/WIAS.PREPRINT.2720

Abstract

In this paper, we extend our study of mass transport in multicomponent isothermal fluids to the incompressible case. For a mixture, incompressibility is defined as the independence of average volume on pressure, and a weighted sum of the partial mass densities of the species stays constant. In this type of models, non solenoidal effects affect the velocity field in the Navier--Stokes equations and, due to different specific volumes of the species, the pressure remains connected to the densities by algebraic formula. By means of a change of variables in the transport problem, we equivalently reformulate the PDE system as to eliminate positivity and incompressibility constraints affecting the density, and prove two type of results: the local-in-time well-posedness in classes of strong solutions, and the global-in-time existence of solutions for initial data sufficiently close to a smooth equilibrium solution.

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WIAS Preprint No. 2148, (2020)

Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth


Authors

  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604
  • Tröltzsch, Fredi

2010 Mathematics Subject Classification

  • 49J20 49K20 49K40 35K57 37N25

Keywords

  • Sparse optimal control, tumor growth models, singular potentials, optimality conditions

DOI

10.20347/WIAS.PREPRINT.2721

Abstract

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions.

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WIAS Preprint No. 2148, (2020)

Pricing options under rough volatility with backward SPDEs


Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Qiu, Jinniao
  • Yao, Yao

2010 Mathematics Subject Classification

  • 91G20 60H15 91G60

Keywords

  • Rough volatility, option pricing, stochastic partial differential equation, machine learning, stochastic Feynman-Kac formula

DOI

10.20347/WIAS.PREPRINT.2745

Abstract

In this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of weak solutions is proved for general nonlinear BSPDEs with unbounded random leading coefficients whose connections with certain forward-backward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep learning-based method is also investigated for the numerical approximations to such BSPDEs and associated non-Markovian pricing problems. Finally, the examples of rough Bergomi type are numerically computed for both European and American options.

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WIAS Preprint No. 2148, (2020)

Micro- and nano-scale engineering and structures shape architecture at nucleation from In-As-Sb-P composition liquid phase on an InAs(100) surface


Authors

  • Gambaryan, Karen M.
  • Marquardt, Oliver
    ORCID: 0000-0002-6642-8988
  • Boeck, Torsten
  • Trampert, Achim

2010 Physics and Astronomy Classification Scheme

  • 73.21.La 73.22.Dj 81.10.Dn

Keywords

  • Nanostructures, nanoarchitecture, liquid phase epitaxy, electronic properties, infrared photodetectors

DOI

10.20347/WIAS.PREPRINT.2775

Abstract

In this review paper we present results of the growth, characterization and electronic properties of In(As,Sb,P) composition strain-induced micro- and nanostructures. Nucleation is performed from In-As-Sb-P quaternary composition liquid phase in Stranski--Krastanow growth mode using steady-state liquid phase epitaxy. Growth features and the shape transformation of pyramidal islands, lens-shape and ellipsoidal type-II quantum dots (QDs), quantum rings and QD-molecules are under consideration. It is shown that the application of a quaternary In(As,Sb,P) composition wetting layer allows not only more flexible control of lattice-mismatch between the wetting layer and an InAs(100) substrate, but also opens up new possibilities for nanoscale engineering and nanoarchitecture of several types of nanostructures. HR-SEM, AFM, TEM and STM are used for nanostructure characterization. Optoelectronic properties of the grown structures are investigated by FTIR and photoresponse spectra measurements. Using an eight-band $mathbfkcdotmathbfp$ model taking strain and built-in electrostatic potentials into account, the electronic properties of a wide range of InAs$_1-x-y$Sb$_x$P$_y$ QDs and QD-molecules are computed. Two types of QDs mid-infrared photodetectors are fabricated and investigated. It is shown that the incorporation of QDs allows to improve some output device characteristics, in particularly sensitivity, and to broaden the spectral range.

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WIAS Preprint No. 2148, (2020)

On the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities


Authors

  • Alphonse, Amal
    ORCID: 0000-0001-7616-3293
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Rautenberg, Carlos N.
    ORCID: 0000-0001-9497-9296

2010 Mathematics Subject Classification

  • 47J20 49J40 49J52 49J50

Keywords

  • Quasi-variational inequality, obstacle problem, directional differentiability, minimal and maximal solutions, ordered solutions

DOI

10.20347/WIAS.PREPRINT.2758

Abstract

In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasi-variational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities.

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WIAS Preprint No. 2148, (2020)

A kinetic model of a polyelectrolyte gel undergoing phase separation


Authors

  • Celora, Giulia L.
  • Hennessy, Matthew G.
  • Münch, Andreas
  • Wagner, Barbara
    ORCID: 0000-0001-8306-3645
  • Waters, Sarah L.

2010 Mathematics Subject Classification

  • 74A30 80A22 34B15

2010 Physics and Astronomy Classification Scheme

  • 83.80.Rs 83.10.Tv

Keywords

  • Polyelectrolyte gel, phase separation, non-equilibrium thermodynamics

DOI

10.20347/WIAS.PREPRINT.2802

Abstract

In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a thermodynamically consistent model for the salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, as well as finite elasticity and multi-component transport. The fully time-dependent model describes the evolution of small changes in the mobile ion concentrations and follows their impact on the large-scale solvent flux and the emergence of long-time pattern formation in the gel. We observe a strong acceleration of the evolution of the free surface when the volume phase transition sets in, as well as the triggering of spinodal decomposition that leads to strong inhomogeneities in the lateral stresses, potentially leading to experimentally visible patterns.

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WIAS Preprint No. 2148, (2020)

Statistical inference for Bures--Wasserstein barycenters


Authors

  • Kroshnin, Alexey
  • Spokoiny, Vladimir
    ORCID: 0000-0002-2040-3427
  • Suvorikova, Alexandra
    ORCID: 0000-0001-9115-7449

2010 Mathematics Subject Classification

  • 60F05

Keywords

  • Bures-Wasserstein barycenters, central limit theorem, optimal transport

DOI

10.20347/WIAS.PREPRINT.2788

Abstract

In this work we introduce the concept of Bures--Wasserstein barycenter $Q_*$, that is essentially a Fréchet mean of some distribution $P$ supported on a subspace of positive semi-definite $d$-dimensional Hermitian operators $H_+(d)$. We allow a barycenter to be constrained to some affine subspace of $H_+(d)$, and we provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of $Q_*$ in both Frobenius norm and Bures--Wasserstein distance, and explain, how the obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.

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WIAS Preprint No. 2148, (2020)

Optimization with learning-informed differential equation constraints and its applications


Authors

  • Dong, Guozhi
    ORCID: 0000-0002-9674-6143
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Papafitsoros, Kostas
    ORCID: 0000-0001-9691-4576

2010 Mathematics Subject Classification

  • 49M15 65J15 65J20 65K10 90C30 35J61 68T07

Keywords

  • PDE constrained optimization, artificial neural network, semilinear PDEs, integrated physicsbased, imaging, learning-informed model, quantitative MRI, semi-smooth Newton SQP algorithm

DOI

10.20347/WIAS.PREPRINT.2754

Abstract

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.

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WIAS Preprint No. 2148, (2020)

Reinforced optimal control


Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Belomestny, Denis
  • Hager, Paul
  • Pigato, Paolo
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266
  • Spokoiny, Vladimir
    ORCID: 0000-0002-2040-3427

2010 Mathematics Subject Classification

  • 91G20 93E24

Keywords

  • Reinforced regression, least squares Monte Carlo, stochastic optimal control

DOI

10.20347/WIAS.PREPRINT.2792

Abstract

Least squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay, Commun. Math. Sci., 18(1):109?121, 2020] proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method?s efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.

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WIAS Preprint No. 2148, (2020)

Generalized self-concordant Hessian-barrier algorithms


Authors

  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Staudigl, Mathias
  • Uribe , Casar A.

2010 Mathematics Subject Classification

  • 90C30 68Q25 65K05

Keywords

  • Non-convex optimization, Bregman divergence, generalized self-concordance, linear constraints

DOI

10.20347/WIAS.PREPRINT.2693

Abstract

Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized selfconcordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify worst-case optimal iteration complexity of the method. Applications in non-convex statistical estimation and Lp-minimization are discussed to given the efficiency of the method.

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WIAS Preprint No. 2148, (2020)

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes


Authors

  • Apel, Thomas
  • Kempf, Volker
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 65N30 65N15 65D05

2010 Physics and Astronomy Classification Scheme

  • 47.10.ad 47.11.Fg

Keywords

  • Anisotropic finite elements, incompressible Navier--Stokes equations, divergence-free methods, pressure-robustness

DOI

10.20347/WIAS.PREPRINT.2702

Abstract

Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case.

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WIAS Preprint No. 2148, (2020)

Near-optimal tensor methods for minimizing gradient norm


Authors

  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Gasnikov, Alexander
  • Ostroukhov, Petr
  • Uribe, A. Cesar
  • Ivanova, Anastasiya

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25

Keywords

  • Convex optimization, tensor methods, gradient norm, nearly optimal methods

DOI

10.20347/WIAS.PREPRINT.2694

Abstract

Motivated by convex problems with linear constraints and, in particular, by entropy-regularized optimal transport, we consider the problem of finding approximate stationary points, i.e. points with the norm of the objective gradient less than small error, of convex functions with Lipschitz p-th order derivatives. Lower complexity bounds for this problem were recently proposed in [Grapiglia and Nesterov, arXiv:1907.07053]. However, the methods presented in the same paper do not have optimal complexity bounds. We propose two optimal up to logarithmic factors methods with complexity bounds with respect to the initial objective residual and the distance between the starting point and solution respectively

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WIAS Preprint No. 2148, (2020)

Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem


Authors

  • Frerichs, Derk
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 65N12 65N30 76D07 76D05 76M10

2010 Physics and Astronomy Classification Scheme

  • 47.10.ad 47.11.Fg

Keywords

  • Incompressible Navier--Stokes equations, mixed virtual element method, pressure-robustness, divergence-free velocity reconstruction, polygonal meshes

DOI

10.20347/WIAS.PREPRINT.2683

Abstract

Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L2 -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.

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WIAS Preprint No. 2148, (2020)

Advances in low-memory subgradient optimization


Authors

  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Gasnikov, Alexander
  • Nurminski, Evgeni
  • Stonyakin, Fedor

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25

Keywords

  • Convex optimization, composite optimization, universal method, mirror prox, acceleration, non-smooth optimization

DOI

10.20347/WIAS.PREPRINT.2676

Abstract

One of the main goals in the development of non-smooth optimization is to cope with high dimensional problems by decomposition, duality or Lagrangian relaxation which greatly reduces the number of variables at the cost of worsening differentiability of objective or constraints. Small or medium dimensionality of resulting non-smooth problems allows to use bundle-type algorithms to achieve higher rates of convergence and obtain higher accuracy, which of course came at the cost of additional memory requirements, typically of the order of n2, where n is the number of variables of non-smooth problem. However with the rapid development of more and more sophisticated models in industry, economy, finance, et all such memory requirements are becoming too hard to satisfy. It raised the interest in subgradient-based low-memory algorithms and later developments in this area significantly improved over their early variants still preserving O(n) memory requirements. To review these developments this chapter is devoted to the black-box subgradient algorithms with the minimal requirements for the storage of auxiliary results, which are necessary to execute these algorithms. To provide historical perspective this survey starts with the original result of N.Z. Shor which opened this field with the application to the classical transportation problem. The theoretical complexity bounds for smooth and non-smooth convex and quasi-convex optimization problems are briefly exposed in what follows to introduce to the relevant fundamentals of non-smooth optimization. Special attention in this section is given to the adaptive step-size policy which aims to attain lowest complexity bounds. Unfortunately the non-differentiability of objective function in convex optimization essentially slows down the theoretical low bounds for the rate of convergence in subgradient optimization compared to the smooth case but there are different modern techniques that allow to solve non-smooth convex optimization problems faster then dictate lower complexity bounds. In this work the particular attention is given to Nesterov smoothing technique, Nesterov Universal approach, and Legendre (saddle point) representation approach. The new results on Universal Mirror Prox algorithms represent the original parts of the survey. To demonstrate application of non-smooth convex optimization algorithms for solution of huge-scale extremal problems we consider convex optimization problems with non-smooth functional constraints and propose two adaptive Mirror Descent methods. The first method is of primal-dual variety and proved to be optimal in terms of lower oracle bounds for the class of Lipschitz-continuous convex objective and constraints. The advantages of application of this method to sparse Truss Topology Design problem are discussed in certain details. The second method can be applied for solution of convex and quasi-convex optimization problems and is optimal in a sense of complexity bounds. The conclusion part of the survey contains the important references that characterize recent developments of non-smooth convex optimization.

Appeared in

  • Numerical Nonsmooth Optimization, A.M. Bagirov, M. Gaudioso, N. Karmitsa, M.M. Mäkelä, S. Taheri, eds., Springer International Publishing, Cham, 2020, pp. 19--59, DOI 10.1007/978-3-030-34910-3_2 .

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WIAS Preprint No. 2148, (2020)

Locking free and gradient robust H(div)-conforming HDG methods for linear elasticity


Authors

  • Fu, Guosheng
  • Lehrenfeld, Christoph
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Streckenbach, Timo
    ORCID: 0009-0001-0874-0463

2010 Mathematics Subject Classification

  • 65N30 65N12 74B05 76D07

Keywords

  • Linear elasticity, nearly incompressible, locking phenomenon, volume-locking, gradient-robustness, discontinuous Galerkin, H(div)-conforming HDG methods

DOI

10.20347/WIAS.PREPRINT.2680

Abstract

Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

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WIAS Preprint No. 2148, (2020)

On the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation


Authors

  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412
  • Maltsi, Anieza
    ORCID: 0000-0003-2417-8770
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 35J10 74J20

Keywords

  • Transmission electron microscopy, electronic Schrödinger equation, elastic scattering, Ewald sphere, dual lattice, Hamiltonian systems, error estimates

DOI

10.20347/WIAS.PREPRINT.2801

Abstract

The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.

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WIAS Preprint No. 2148, (2020)

Further regularity and uniqueness results for a non-isothermal Cahn--Hilliard equation


Authors

  • Ipocoana, Erica
  • Zafferi, Andrea
    ORCID: 0000-0003-4012-7666

2010 Mathematics Subject Classification

  • 35Q35 35D35 80A22 35A02 35B65

Keywords

  • Cahn--Hilliard, non-isothermal model, regularity of solutions, uniqueness

DOI

10.20347/WIAS.PREPRINT.2716

Abstract

The aim of this paper is to establish new regularity results for a non-isothermal Cahn--Hilliard system in the two-dimensional setting. The main achievement is a crucial L∞ estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.

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WIAS Preprint No. 2148, (2020)

On the spatially asymptotic structure of time-periodic solutions to the Navier--Stokes equations


Authors

  • Eiter, Thomas
    ORCID: 0000-0002-7807-1349

2010 Mathematics Subject Classification

  • 35Q30 35B10 35C20 76D05 35E05

Keywords

  • Navier--Stokes, time-periodic solutions, asymptotic expansion, Oseen system, fundamental solution

DOI

10.20347/WIAS.PREPRINT.2727

Abstract

The asymptotic behavior of weak time-periodic solutions to the Navier--Stokes equations with a drift term in the three-dimensional whole space is investigated. The velocity field is decomposed into a time-independent and a remaining part, and separate asymptotic expansions are derived for both parts and their gradients. One observes that the behavior at spatial infinity is determined by the corresponding Oseen fundamental solutions.

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WIAS Preprint No. 2148, (2020)

Spinodal decomposition and collapse of a polyelectrolyte gel


Authors

  • Celora, Giulia L.
  • Hennessy, Matthew G.
  • Münch, Andreas
  • Waters, Sarah L.
  • Wagner, Barbara
    ORCID: 0000-0001-8306-3645

2010 Mathematics Subject Classification

  • 74A30 80A22 34B15

2010 Physics and Astronomy Classification Scheme

  • 83.80.Rs 83.10.Tv.

Keywords

  • Polyelectrolyte gel, phase separation, collapse

DOI

10.20347/WIAS.PREPRINT.2731

Abstract

The collapse of a polyelectrolyte gel in a (monovalent) salt solution is analysed using a new model that includes interfacial gradient energy to account for phase separation in the gel, finite elasticity and multicomponent transport. We carry out a linear stability analysis to determine the stable and unstable spatially homogeneous equilibrium states and how they phase separate into localized regions that eventually coarsen to a new stable state. We then investigate the problem of a collapsing gel as a response to increasing the salt concentration in the bath. A phase space analysis reveals that the collapse is obtained by a front moving through the gel that eventually ends in a new stable equilibrium. For some parameter ranges, these two routes to gel shrinking occur together.

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WIAS Preprint No. 2148, (2020)

Simulating the electronic properties of semiconductor nanostructures using multiband $kcdot p$ models


Authors

  • Marquardt, Oliver
    ORCID: 0000-0002-6642-8988

2010 Physics and Astronomy Classification Scheme

  • 71.15.-m 02.70.-c 81.07.Gf

Keywords

  • Nanowires, quantum dots, electronic properties, $kcdot p$ models

DOI

10.20347/WIAS.PREPRINT.2773

Abstract

The eight-band $kcdot p$ formalism been successfully applied to compute the electronic properties of a wide range of semiconductor nanostructures in the past and can be considered the backbone of modern semiconductor heterostructure modelling. However, emerging novel material systems and heterostructure fabrication techniques raise questions that cannot be answered using this well-established formalism, due to its intrinsic limitations. The present article reviews recent studies on the calculation of electronic properties of semiconductor nanostructures using a generalized multiband $kcdot p$ approach that allows both the application of the eight-band model as well as more sophisticated approaches for novel material systems and heterostructures.

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WIAS Preprint No. 2148, (2020)

Existence of solutions of a finite element flux-corrected-transport scheme


Authors

  • John, Volker
    ORCID: 0000-0002-2711-4409
  • Knobloch, Petr
    ORCID: 0000-0003-2709-5882

2010 Mathematics Subject Classification

  • 65M60

Keywords

  • Evolutionary convection-diffusion-reaction equations, transport equations, finite element method, nonlinear flux-corrected-transport scheme, existence of a solution, Brouwer's fixed-point theorem

DOI

10.20347/WIAS.PREPRINT.2761

Abstract

The existence of a solution is proved for a nonlinear finite element flux-corrected-transport (FEM-FCT) scheme with arbitrary time steps for evolutionary convection-diffusion-reaction equations and transport equations.

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WIAS Preprint No. 2148, (2020)

An effective bulk-surface thermistor model for large-area organic light-emitting diodes


Authors

  • Glitzky, Annegret
    ORCID: 0000-0003-1995-5491
  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Nika, Grigor
    ORCID: 0000-0002-4403-6908

2010 Mathematics Subject Classification

  • 35Q79 35J25 80A20

Keywords

  • Dimension reduced thermistor system, existence of weak solutions, entropy solutions, organic light emitting diode, self-heating

DOI

10.20347/WIAS.PREPRINT.2757

Abstract

The existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of large-area organic light-emitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the three-dimensional bulk glass substrate and two semi-linear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the current-flow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixed-point theorem. Moreover, since the heat sources are a priori only in $L^1$, the concept of entropy solutions is used.

Appeared in

  • Port. Math., 78 (2021), pp. 187--210, DOI 10.4171/PM/2066 under the new title ''Analysis of a bulk-surface thermistor model for large-area organic LEDs" .

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WIAS Preprint No. 2148, (2020)

Modeling and simulation of the lateral photovoltage scanning method


Authors

  • Farrell, Patricio
    ORCID: 0000-0001-9969-6615
  • Kayser, Stefan
  • Rotundo, Nella

2010 Mathematics Subject Classification

  • 35Q81 35K57 65N08

Keywords

  • Lateral-photovoltage-scanning method (LPS), semiconductor simulation, van Roosbroeck system, finite volume simulation, crystal growth

DOI

10.20347/WIAS.PREPRINT.2784

Abstract

The fast, cheap and nondestructive lateral photovoltage scanning (LPS) method detects inhomogeneities in semiconductors crystals. The goal of this paper is to model and simulate this technique for a given doping profile. Our model is based on the semiconductor device equations combined with a nonlinear boundary condition, modelling a volt meter. To validate our 2D and 3D finite volume simulations, we use theory developed by Tauc [21] to derive three analytical predictions which our simulation results corroborate, even for anisotropic 2D and 3D meshes. Our code runs about two orders of magnitudes faster than earlier implementations based on commercial software [15]. It also performs well for small doping concentrations which previously could not be simulated at all due to numerical instabilities. Our simulations provide experimentalists with reference laser powers for which meaningful voltages can still be measured. For higher laser power the screening effect does not allow this anymore.

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WIAS Preprint No. 2148, (2020)

Mode competition in broad-ridge-waveguide lasers


Authors

  • Köster, Jan-Philipp
  • Putz, Alexander
  • Wenzel, Hans
    ORCID: 0000-0003-1726-0223
  • Wünsche, Hans-Jürgen
  • Radziunas, Mindaugas
    ORCID: 0000-0003-0306-1266
  • Stephan, Holger
    ORCID: 0000-0002-6024-5355
  • Wilkens, Martin
  • Zeghuzi, Anissa
  • Knigge, Andrea

2010 Mathematics Subject Classification

  • 78A60 35Q60 78-04 78A50

Keywords

  • High-brightness laser diodes, ridge-waveguide lasers, device simulation, traveling-wave model, modal analysis, beam steering, coherent mode coupling

DOI

10.20347/WIAS.PREPRINT.2764

Abstract

The lateral brightness achievable with high-power GaAs-based laser diodes having long and broad waveguides is commonly regarded to be limited by the onset of higher-order lateral modes. For the study of the lateral-mode competition two complementary simulation tools are applied, representing different classes of approximations. The first tool bases on a completely incoherent superposition of mode intensities and disregards longitudinal effects like spatial hole burning, whereas the second tool relies on a simplified carrier transport and current flow. Both tools yield agreeing power-current characteristics that fit the data measured for 5 to 23 µm wide ridges. Also, a similarly good qualitative conformance of the near and far fields is found. However, the threshold of individual modes, the partition of power between them at a given current, and details of the near and far fields show differences. These differences are the consequence of a high sensitivity of the mode competition to details of the models and of the device structure. Nevertheless, it can be concluded concordantly that the brightness rises with increasing ridge width irrespective of the onset of more and more lateral modes. The lateral brightness 2W · mm¯¹ 1mrad¯¹ at 10MW · cm¯²2 power density on the front facet of the investigated laser with widest ridge (23 µm) is comparable with best values known from much wider broad-area lasers. In addition, we show that one of the simulation tools is able to predict beam steering and coherent beam

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WIAS Preprint No. 2148, (2020)

Detecting striations via the lateral photovoltage scanning method without screening effect


Authors

  • Kayser, Stefan
  • Farrell, Patricio
    ORCID: 0000-0001-9969-6615
  • Rotundo, Nella

2010 Mathematics Subject Classification

  • 35Q81 35K57 65N08

Keywords

  • Lateral-photovoltage-scanning method (LPS), semiconductor simulation, van Roosbroeck system, finite volume simulation, crystal growth

DOI

10.20347/WIAS.PREPRINT.2785

Abstract

The lateral photovoltage scanning method (LPS) detects doping inhomogeneities in semiconductors such as Si, Ge and Si(x)Ge(1-x) in a cheap, fast and nondestructive manner. LPS relies on the bulk photovoltaic effect and thus can detect any physical quantity affecting the band profiles of the sample. LPS finite volume simulation using commercial software suffer from long simulation times and convergence instabilities. We present here an open-source finite volume simulation for a 2D Si sample using the ddfermi simulator. For low injection conditions we show that the LPS voltage is proportional to the doping gradient as previous theory suggested under certain conditions. For higher injection conditions we directly show how the LPS voltage and the doping gradient differ and link the physical effect of lower local resolution to the screening effect. Previously, the loss of local resolution was assumed to be only connected to the enlargement of the excess charge carrier distribution.

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WIAS Preprint No. 2148, (2020)

Dynamic probabilistic constraints under continuous random distributions


Authors

  • González Grandón, Tatiana
  • Henrion, René
    ORCID: 0000-0001-5572-7213
  • Pérez-Aros, Pedro

2010 Mathematics Subject Classification

  • 90C15 49K45

Keywords

  • Dynamic probabilistic constraints, chance constraints, continuous distributions, decision rules, stochastic programming

DOI

10.20347/WIAS.PREPRINT.2783

Abstract

The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.

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WIAS Preprint No. 2148, (2020)

Generalized Poisson--Nernst--Planck-based physical model of O$_2$ I LSM I YSZ electrode


Authors

  • Miloš, Vojtěch
  • Vágner, Petr
    ORCID: 0000-0001-5952-0025
  • Budáč, Daniel
  • Carda, Michal
  • Paidar, Martin
  • Fuhrmann, Jürgen
    ORCID: 0000-0003-4432-2434
  • Bouzek, Karel

2010 Mathematics Subject Classification

  • 65N30 78A57 80A17

Keywords

  • Solid oxide cells, charge layer, thermodynamics, electrode potentia

DOI

10.20347/WIAS.PREPRINT.2797

Abstract

The paper presents a generalized Poisson-Nernst-Planck model of an yttria-stabilized zirconia electrolyte developed from first principles of nonequilibrium thermodynamics which allows for spatial resolution of the space charge layer. It takes into account limitations in oxide ion concentrations due to the limited availability of oxygen vacancies. The electrolyte model is coupled with a reaction kinetic model describing the triple phase boundary with electron conducting lanthanum strontium manganite and gaseous phase oxygen. By comparing the outcome of numerical simulations based on different formulations of the kinetic equations with results of EIS and CV measurements we attempt to discern the existence of separate surface lattice sites for oxygen adatoms and O2- from the assumption of shared ones. Furthermore, we discern mass-action kinetics models from exponential kinetics models.

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WIAS Preprint No. 2148, (2020)

Uncertainty quantification in image segmentation using the Ambrosio--Tortorelli approximation of the Mumford--Shah energy


Authors

  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Stengl, Steven-Marian
  • Surowiec, Thomas M.
    ORCID: 0000-0003-2473-4984

2010 Mathematics Subject Classification

  • 62D99 65N75 68U10 65K10

Keywords

  • Image segmentation, Mumford-Shah, Ambrosio-Tortorelli, measurable selection, Monte-Carlo, sampling

DOI

10.20347/WIAS.PREPRINT.2703

Abstract

The quantification of uncertainties in image segmentation based on the Mumford-Shah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the Ambrosio-Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods and the limitations of other popular methods. Numerical examples illustrate the theoretical findings.

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WIAS Preprint No. 2148, (2020)

An enumerative formula for the spherical cap discrepancy


Authors

  • Heitsch, Holger
    ORCID: 0000-0002-2692-4602
  • Henrion, René
    ORCID: 0000-0001-5572-7213

2010 Mathematics Subject Classification

  • 11K38 90C15

Keywords

  • Spherical cap discrepancy, uniform distribution on sphere, optimality conditions

DOI

10.20347/WIAS.PREPRINT.2744

Abstract

The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.

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WIAS Preprint No. 2148, (2020)

Assessing the quality of the excess chemical potential flux scheme for degenerate semiconductor device simulation


Authors

  • Abdel, Dilara
    ORCID: 0000-0003-3477-7881
  • Farrell, Patricio
    ORCID: 0000-0001-9969-6615
  • Fuhrmann, Jürgen
    ORCID: 0000-0003-4432-2434

2010 Mathematics Subject Classification

  • 35Q81 35K57 65N08

Keywords

  • degenerate semiconductors, drift-diffusion equations, finite volume method, flux discretization, Scharfetter--Gummel scheme

DOI

10.20347/WIAS.PREPRINT.2787

Abstract

The van Roosbroeck system models current flows in (non-)degenerate semiconductor devices. Focusing on the stationary model, we compare the excess chemical potential discretization scheme, a flux approximation which is based on a modification of the drift term in the current densities, with another state-of-the-art Scharfetter-Gummel scheme, namely the diffusion-enhanced scheme. Physically, the diffusion-enhanced scheme can be interpreted as a flux approximation which modifies the thermal voltage. As a reference solution we consider an implicitly defined integral flux, using Blakemore statistics. The integral flux refers to the exact solution of a local two point boundary value problem for the continuous current density and can be interpreted as a generalized Scharfetter-Gummel scheme. All numerical discretization schemes can be used within a Voronoi finite volume method to simulate charge transport in (non-)degenerate semiconductor devices. The investigation includes the analysis of Taylor expansions, a derivation of error estimates and a visualization of errors in local flux approximations to extend previous discussions. Additionally, drift-diffusion simulations of a p-i-n device are performed.

Appeared in

  • Opt. Quantum Electron., 53 (2021), pp. 163/1--163/10.

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WIAS Preprint No. 2148, (2020)

Discrete approximation of dynamic phase-field fracture in visco-elastic materials


Authors

  • Thomas, Marita
    ORCID: 0000-0001-9172-014X
  • Tornquist, Sven

2010 Mathematics Subject Classification

  • 74H10 74H20 74H30 35M86 35Q74

Keywords

  • Visco-elastodynamic damage, Ambrosio-Tortorelli model for phase-field fracture, viscous evolution, rate-independent limit, temporal regularity of solutions

DOI

10.20347/WIAS.PREPRINT.2798

Abstract

This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively 1-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

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WIAS Preprint No. 2148, (2020)

Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis


Authors

  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Signori, Andrea
    ORCID: 0000-0001-7025-977X
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 49J20 49K20 49K40 35K57 37N2

Keywords

  • Optimal control, tumor growth models, singular potentials, optimality conditions, second-order analysis

DOI

10.20347/WIAS.PREPRINT.2770

Abstract

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn--Hilliard type including chemotaxis with possibly singular anpotentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

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WIAS Preprint No. 2148, (2020)

Nonparametric change point detection in regression


Authors

  • Avanesov, Valeriy

2010 Mathematics Subject Classification

  • 62M10 62H15

Keywords

  • Bootstrap, change point detection, nonparametrics, regression, multiscale

DOI

10.20347/WIAS.PREPRINT.2687

Abstract

This paper considers the prominent problem of change-point detection in regression. The study suggests a novel testing procedure featuring a fully data-driven calibration scheme. The method is essentially a black box, requiring no tuning from the practitioner. The approach is investigated from both theoretical and practical points of view. The theoretical study demonstrates proper control of first-type error rate under H0 and power approaching 1 under H1. The experiments conducted on synthetic data fully support the theoretical claims. In conclusion, the method is applied to financial data, where it detects sensible change-points. Techniques for change-point localization are also suggested and investigated

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WIAS Preprint No. 2148, (2020)

Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations


Authors

  • Lederer, Philip Lukas
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 65N15 65N30 76D07 76M10

Keywords

  • Incompressible Navier-Stokes equations, mixed finite elements, pressure-robustness, a posteriori error estimators, equilibrated fluxes, adaptive mesh refinement

DOI

10.20347/WIAS.PREPRINT.2750

Abstract

This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1.

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WIAS Preprint No. 2148, (2020)

On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell--Stefan and the Fick--Onsager approach


Authors

  • Bothe, Dieter
  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500

2010 Mathematics Subject Classification

  • 76R50 76T30 80A20 80A17 35K57 76V05 80A32 92E20

Keywords

  • Multicomponent diffusion, irreversible thermodynamics, entropy production, Maxwell-Stefan diffusivities, core-diagonal closure, Darken equation, cross-diffusion, sign of diffusivities

DOI

10.20347/WIAS.PREPRINT.2749

Abstract

This paper revisits the modeling of multicomponent diffusion within the framework of thermodynamics of irreversible processes. We briefly review the two well-known main approaches, leading to the generalized Fick--Onsager multicomponent diffusion fluxes or to the generalized Maxwell--Stefan equations. The latter approach has the advantage that the resulting fluxes are consistent with non-negativity of the partial mass densities for non-singular and non-degenerate Maxwell--Stefan diffusivities. On the other hand, this approach requires computationally expensive matrix inversions since the fluxes are only implicitly given. We propose and discuss a novel and more direct closure which avoids the inversion of the Maxwell--Stefan equations. It is shown that all three closures are actually equivalent under the natural requirement of positivity for the concentrations, thus revealing the general structure of continuum thermodynamical diffusion fluxes.

Appeared in

  • Int. J. Engineering Sci., vol. 184, March 2023, 103818.

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WIAS Preprint No. 2148, (2020)

Fully discrete approximation of rate-independent damage models with gradient regularization


Authors

  • Bartels, Sören
  • Milicevic, Marijo
  • Thomas, Marita
    ORCID: 0000-0001-9172-014X
  • Weber, Nico

2010 Mathematics Subject Classification

  • 35K86 74R05 49J45 49S05 65M60 65M12

Keywords

  • Partial damage, damage evolution with gradient regularization, semistable energetic solutions, numerical approximation, iterative solution

DOI

10.20347/WIAS.PREPRINT.2707

Abstract

This work provides a convergence analysis of a time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient regularization as well as a non-smooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rate-independent gradient damage is based on a Variable ADMM-method to approximate the nonsmooth contribution. Space-discretization is based on P1 finite elements and the algorithm directly couples the time-step size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BV-functions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method.

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WIAS Preprint No. 2148, (2020)

Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization


Authors

  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Papafitsoros, Kostas
    ORCID: 0000-0001-9691-4576
  • Rautenberg, Carlos N.
    ORCID: 0000-0001-9497-9296
  • Sun, Hongpeng

2010 Mathematics Subject Classification

  • 94A08 68U10 49K20 49Q20 65K15 26A45

Keywords

  • Image restoration, image denoising, total generalized variation, spatially distributed regularization, weight, bilevel optimization

DOI

10.20347/WIAS.PREPRINT.2689

Abstract

Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.

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WIAS Preprint No. 2148, (2020)

Oracle complexity separation in convex optimization


Authors

  • Ivanova, Anastasiya
  • Gasnikov, Alexander
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Dvinskikh, Darina
  • Tyurin, Alexander
  • Vorontsova, Evgeniya
  • Pasechnyuk, Dmitry

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25 65K15

Keywords

  • Convex optimization, composite optimization, proximal method, acceleration, random coordinate descent, variance reduction

DOI

10.20347/WIAS.PREPRINT.2711

Abstract

Ubiquitous in machine learning regularized empirical risk minimization problems are often composed of several blocks which can be treated using different types of oracles, e.g., full gradient, stochastic gradient or coordinate derivative. Optimal oracle complexity is known and achievable separately for the full gradient case, the stochastic gradient case, etc. We propose a generic framework to combine optimal algorithms for different types of oracles in order to achieve separate optimal oracle complexity for each block, i.e. for each block the corresponding oracle is called the optimal number of times for a given accuracy. As a particular example, we demonstrate that for a combination of a full gradient oracle and either a stochastic gradient oracle or a coordinate descent oracle our approach leads to the optimal number of oracle calls separately for the full gradient part and the stochastic/coordinate descent part.

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WIAS Preprint No. 2148, (2020)

Mesh generation for periodic 3D microstructure models and computation of effective properties


Authors

  • Landstorfer, Manuel
    ORCID: 0000-0002-0565-2601
  • Prifling, Benedikt
  • Schmidt, Volker

2010 Mathematics Subject Classification

  • 35C20 65N50 35B27 60D05 74N15

Keywords

  • Mesh generation, porous material modeling, stochastic microstructure modeling, effective properties, spherical harmonics, periodic homogenization.

DOI

10.20347/WIAS.PREPRINT.2738

Abstract

Understanding and optimizing effective properties of porous functional materials, such as permeability or conductivity, is one of the main goals of materials science research with numerous applications. For this purpose, understanding the underlying 3D microstructure is crucial since it is well known that the materials? morphology has an significant impact on their effective properties. Because tomographic imaging is expensive in time and costs, stochastic microstructure modeling is a valuable tool for virtual materials testing, where a large number of realistic 3D microstructures can be generated and used as geometry input for spatially-resolved numerical simulations. Since the vast majority of numerical simulations is based on solving differential equations, it is essential to have fast and robust methods for generating high-quality volume meshes for the geometrically complex microstructure domains. The present paper introduces a novel method for generating volume-meshes with periodic boundary conditions based on an analytical representation of the 3D microstructure using spherical harmonics. Due to its generality, the present method is applicable to many scientific areas. In particular, we present some numerical examples with applications to battery research by making use of an already existing stochastic 3D microstructure model that has been calibrated to eight differently compacted cathodes.

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WIAS Preprint No. 2148, (2020)

Stability of deep neural networks via discrete rough paths


Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Friz, Peter
    ORCID: 0000-0003-2571-8388
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2010 Mathematics Subject Classification

  • 60L10 60L70 60L90

Keywords

  • Deep neural networks, iterated sums signature

DOI

10.20347/WIAS.PREPRINT.2732

Abstract

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks. In particular we derive stability bounds in terms of the total p-variation of trained weights for any p ≥ 1.

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WIAS Preprint No. 2148, (2020)

Site-monotonicity properties for reflection positive measures with applications to quantum spin systems


Authors

  • Lees, Benjamin
  • Taggi, Lorenzo

2010 Mathematics Subject Classification

  • 82B20 82B26 82B41 05A05

Keywords

  • Quantum spin systems, random loop models, infrared bound, phase transitions

DOI

10.20347/WIAS.PREPRINT.2713

Abstract

We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application of such a general theorem, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that such spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates -- improving previous positivity results which hold for the Cesàro sum -- and we derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model, lattice permutations, thus extending the previous results of Lees and Taggi (2019).

Appeared in

  • J. Stat. Phys., 183 (2021), art. 38.

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WIAS Preprint No. 2148, (2020)

Optimality conditions for convex stochastic optimization problems in Banach spaces with almost sure state constraint


Authors

  • Geiersbach, Caroline
    ORCID: 0000-0002-6518-7756
  • Wollner, Winnifried

2010 Mathematics Subject Classification

  • 49K20 49K21 49K45 49N15 49J53

Keywords

  • PDE-constrained optimization under uncertainty, optimization in Banach spaces, optimality conditions, convex stochastic optimization in Banach spaces, two-stage stochastic optimization, regular Lagrange multipliers, duality

DOI

10.20347/WIAS.PREPRINT.2755

Abstract

We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty.

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WIAS Preprint No. 2148, (2020)

A rigorous derivation and energetics of a wave equation with fractional damping


Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Netz, Roland R.
  • Zendehroud, Sina
    ORCID: 0000-0002-5822-722X

2010 Mathematics Subject Classification

  • 35Q35 35Q74 35R11 74J15 74K15

Keywords

  • Bulk-interface coupling, surface waves, energy-dissipation balance, fractional derivatives, convergence of semigroups, parabolic Dirichlet-to-Neumann map, dispersion relation, damping of order 3/2

DOI

10.20347/WIAS.PREPRINT.2718

Abstract

We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2.

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WIAS Preprint No. 2148, (2020)

Relating a rate-independent system and a gradient system for the case of one-homogeneous potentials


Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 37L05 47H20 47J35 47J40

Keywords

  • Gradient flow energetic solutions, rate-independent system, contraction semigroup, set of stable states, time parametrization

DOI

10.20347/WIAS.PREPRINT.2771

Abstract

We consider a non-negative and one-homogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent functional $mathcal E(t,u)=t mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

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WIAS Preprint No. 2148, (2020)

Distribution of cracks in a chain of atoms at low temperature


Authors

  • Jansen, Sabine
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Schmidt, Bernd
  • Theil, Florian

2010 Mathematics Subject Classification

  • 82B21 74B20 74G65 60F10

Keywords

  • Equilibrium statistical mechanics, atomistic models of elasticity, fracture, lattice gas of defects

DOI

10.20347/WIAS.PREPRINT.2789

Abstract

We consider a one-dimensional classical many-body system with interaction potential of Lennard--Jones type in the thermodynamic limit at low temperature 1/β ∈ (0, ∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of N exp(-β e surf /2) with e surf > 0 a surface energy.

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WIAS Preprint No. 2148, (2020)

Numerical analysis for nematic electrolytes


Authors

  • Baňas, L'ubomír
  • Lasarzik, Robert
    ORCID: 0000-0002-1677-6925
  • Prohl, Andreas

2010 Mathematics Subject Classification

  • 35Q35 35Q70 65M60 74E10

Keywords

  • Existence, approximation, Navier-Stokes, Ericksen-Leslie, Nernst-Planck, nematic electrolytes, finite element method, fully discrete scheme, convergence analysis

DOI

10.20347/WIAS.PREPRINT.2717

Abstract

We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.

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WIAS Preprint No. 2148, (2020)

Additive functionals as rough paths


Authors

  • Deuschel, Jean-Dominique
  • Orenshtein, Tal
  • Perkowski, Nicolas

Keywords

  • Rough paths, invariance principles in the rough, path topology, additive functionals of Markov processes, Kipnis--Varadhan theory, homogenization, random conductance model, random walks with random conductances

DOI

10.20347/WIAS.PREPRINT.2685

Abstract

We consider additive functionals of stationary Markov processes and show that under Kipnis--Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (non-reversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a non-reversible Ornstein-Uhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step we prove an estimate for the p-variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path Burkholder-Davis-Gundy inequality for local martingale rough paths of [FV08], [CF19] and [FZ18] to the case where only the integrator is a local martingale.

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WIAS Preprint No. 2148, (2020)

Alternating minimization methods for strongly convex optimization


Authors

  • Tupitsa, Nazarii
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Gasnikov, Alexander
  • Guminov, Sergey

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25

Keywords

  • Convex optimization, alternating minimization, block-coordinate method, complexity analysis

DOI

10.20347/WIAS.PREPRINT.2692

Abstract

We consider alternating minimization procedures for convex optimization problems with variable divided in many block, each block being amenable for minimization with respect to its variable with freezed other variables blocks. In the case of two blocks, we prove a linear convergence rate for alternating minimization procedure under Polyak-Łojasiewicz condition, which can be seen as a relaxation of the strong convexity assumption. Under strong convexity assumption in many-blocks setting we provide an accelerated alternating minimization procedure with linear rate depending on the square root of the condition number as opposed to condition number for the non-accelerated method.

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WIAS Preprint No. 2148, (2020)

Convergence bounds for empirical nonlinear least-squares


Authors

  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Trunschke, Philipp
  • Schneider, Reinhold

2010 Mathematics Subject Classification

  • 41A10 41A25 41A65 62E17 93E24

Keywords

  • Multivariate approximation, restricted isometry property, weighted least squares, tensor representation, convergence rates, error analysis, nonlinear approximation, conditional sampling

DOI

10.20347/WIAS.PREPRINT.2714

Abstract

We consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds.

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WIAS Preprint No. 2148, (2020)

Branching random walks in random environment: A survey


Authors

  • König, Wolfgang
    ORCID: 0000-0002-7673-4364

2010 Mathematics Subject Classification

  • 60J80 60J55 60F10 60K37

Keywords

  • Multitype branching random walk, random potential, parabolic Anderson model, Feynman--Kac-type formula, annealed moments, large deviations

DOI

10.20347/WIAS.PREPRINT.2779

Abstract

We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.

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WIAS Preprint No. 2148, (2020)

Low-dimensional approximations of high-dimensional asset price models


Authors

  • Redmann, Martin
    ORCID: 0000-0001-5182-9773
  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Goyal, Pawan

2010 Mathematics Subject Classification

  • 91G20 91G60 93A15 60H10 65D32

Keywords

  • Model order reduction, Black Scholes model, Heston model, option pricing

DOI

10.20347/WIAS.PREPRINT.2706

Abstract

We consider high-dimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have non-zero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out.

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WIAS Preprint No. 2148, (2020)

EDP-convergence for nonlinear fast-slow reaction systems with detailed balance


Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Peletier, Mark A.
    ORCID: 0000-0001-9663-3694
  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2010 Mathematics Subject Classification

  • 49S05 47J30 92E20 34E13

Keywords

  • Nonlinear reaction system with detailed balance, fast-reaction limit, gradient structure, gradient system, EDP-convergence, energy-dissipation principle, Gamma-convergence

DOI

10.20347/WIAS.PREPRINT.2781

Abstract

We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.

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WIAS Preprint No. 2148, (2020)

Gradient methods for problems with inexact model of the objective


Authors

  • Stonyakin, Fedor
  • Dvinskikh, Darina
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343
  • Kroshnin, Alexey
  • Kuznetsova, Olesya
  • Agafonov, Artem
  • Gasnikov, Alexander
  • Tyurin, Alexander
  • Uribe, Cesar A.
  • Pasechnyuk, Dmitry
  • Artamonov, Sergei

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25

Keywords

  • Gradient method, inexact oracle, strong convexity, relative smoothness, Bregman divergence

DOI

10.20347/WIAS.PREPRINT.2688

Abstract

We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments.

Appeared in

  • Proceedings of the 18th International Conference on Mathematical Optimization Theory and Operations Research (MOTOR 2019), M. Khachay, Y. Kochetov, P. Pardalos, eds., vol. 11548 of Lecture Notes in Computer Science, Springer Nature Switzerland AG 2019, Cham, Switzerland, 2019, pp. 97--114, DOI 10.1007/978-3-030-22629-9_8 .

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WIAS Preprint No. 2148, (2020)

On the numerical range of second order elliptic operators with mixed boundary conditions in L$^p$


Authors

  • Chill, Ralph
  • Meinlschmidt, Hannes
    ORCID: 0000-0002-5874-8017
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 35B65 35J15 47A12

Keywords

  • Bidomain system, j-subgradient, gradient system, FitzHugh--Nagumo model

DOI

10.20347/WIAS.PREPRINT.2723

Abstract

We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on Lp in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin- instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions.

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WIAS Preprint No. 2148, (2020)

Optimal decentralized distributed algorithms for stochastic convex optimization


Authors

  • Gorbunov, Eduard
  • Dvinskikh, Darina
  • Gasnikov, Alexander

2010 Mathematics Subject Classification

  • 90C25 90C06 90C90

Keywords

  • Convex optimization, stochastic optimization, primal and dual methods, distributed methods, decentralized algorithms, first-order methods, optimal complexity bounds, mini-batch

DOI

10.20347/WIAS.PREPRINT.2691

Abstract

We consider stochastic convex optimization problems with affine constraints and develop several methods using either primal or dual approach to solve it. In the primal case we use special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm to solve the same problems in the non-smooth case. We prove the convergence guarantees in smooth convex case with deterministic first-order oracle. We propose and analyze three novel methods to handle stochastic convex optimization problems with affine constraints: SPDSTM, R-RRMA-AC-SA and SSTM_sc. All methods use stochastic dual oracle. SPDSTM is the stochastic primal-dual modification of STM and it is applied for the dual problem when the primal functional is strongly convex and Lipschitz continuous on some ball. R-RRMA-AC-SA is an accelerated stochastic method based on the restarts of RRMA-AC-SA and SSTM_sc is just stochastic STM for strongly convex problems. Both methods are applied to the dual problem when the primal functional is strongly convex, smooth and Lipschitz continuous on some ball and use stochastic dual first-order oracle. We develop convergence analysis for these methods for the unbiased and biased oracles respectively. Finally, we apply all aforementioned results and approaches to solve decentralized distributed optimization problem and discuss optimality of the obtained results in terms of communication rounds and number of oracle calls per node.

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WIAS Preprint No. 2148, (2020)

Phase transitions for the Boolean model of continuum percolation for Cox point processes


Authors

  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Tóbiás, András
  • Cali, Eli

2010 Mathematics Subject Classification

  • 82B43 60G55 60K35

Keywords

  • Cox point processes, continuum percolation, random environment, Boolean model, Gilbert disk model, random radii, moments, diameter of cluster, volume of cluster, number of points in cluster, uniqueness of infinite cluster, complete coverage, ergodicity, stabilization, exponential stabilization, polynomial stabilization, b-dependence, essential connectedness, shot-noise fields, Boolean models on Boolean models

DOI

10.20347/WIAS.PREPRINT.2704

Abstract

We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

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WIAS Preprint No. 2148, (2020)

On the optimal combination of tensor optimization methods


Authors

  • Kamzolov, Dmitry
  • Gasnikov, Alexander
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343

2010 Mathematics Subject Classification

  • 90C30 90C25 68Q25 65K15

Keywords

  • Tensor methods, sliding, uniformly convex function, inexactness, Taylor expansion, complexity

DOI

10.20347/WIAS.PREPRINT.2710

Abstract

We consider the minimization problem of a sum of a number of functions having Lipshitz p -th order derivatives with different Lipschitz constants. In this case, to accelerate optimization, we propose a general framework allowing to obtain near-optimal oracle complexity for each function in the sum separately, meaning, in particular, that the oracle for a function with lower Lipschitz constant is called a smaller number of times. As a building block, we extend the current theory of tensor methods and show how to generalize near-optimal tensor methods to work with inexact tensor step. Further, we investigate the situation when the functions in the sum have Lipschitz derivatives of a different order. For this situation, we propose a generic way to separate the oracle complexity between the parts of the sum. Our method is not optimal, which leads to an open problem of the optimal combination of oracles of a different order.

Appeared in

  • Optimization and Applications. OPTIMA 2020, N. Olenev, Y. Evtushenko, M. Khachay, V. Malkova, eds., vol. 12422 of Lecture Notes in Computer Science, Springer International Publishing, Cham, 2020, pp. 166--183, DOI 10.1007/978-3-030-62867-3_13 .

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WIAS Preprint No. 2148, (2020)

Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential


Authors

  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Perkowski, Nicolas
  • van Zuijlen, Willem
    ORCID: 0000-0002-2079-0359

2010 Mathematics Subject Classification

  • 60H17 60H25 60L40 82B4 35J10 35P15

Keywords

  • Parabolic Anderson model, Anderson Hamiltonian, white-noise potential, singular SPDE, paracontrolled distribution, regularization in two dimensions, intermittency, almost-sure large-time asymptotics, principal eigenvalue of random Schrödinger operator

DOI

10.20347/WIAS.PREPRINT.2765

Abstract

We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.

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WIAS Preprint No. 2148, (2020)

Quantitative heat kernel estimates for diffusions with distributional drift


Authors

  • Perkowski, Nicolas
  • van Zuijlen, Willem
    ORCID: 0000-0002-2079-0359

2010 Mathematics Subject Classification

  • 60H10 35A08

Keywords

  • Heat kernel bound, singular diffusion, parametrix method

DOI

10.20347/WIAS.PREPRINT.2768

Abstract

We consider the stochastic differential equation on ℝ d given by d X t = b(t,Xt ) d t + d Bt, where B is a Brownian motion and b is considered to be a distribution of regularity > - 1/2. We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat kernel bounds for Γt with explicit dependence on t and the norm of b.

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WIAS Preprint No. 2148, (2020)

Absence of percolation in graphs based on stationary point processes with degrees bounded by two


Authors

  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Tóbiás, András

2010 Mathematics Subject Classification

  • 82B43 60G55 60K35

Keywords

  • Continuum percolation, stationary point processes, degree bounds, bidirectional k-nearest neighbor graph, edge-preserving property, signal-to-interference ratio

DOI

10.20347/WIAS.PREPRINT.2774

Abstract

We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.

Appeared in

  • Random Structures Algorithms, published online on 30.03.2022 (2022), DOI 10.1002/rsa.21084 .

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WIAS Preprint No. 2148, (2020)

Modeling of chemical reaction systems with detailed balance using gradient structures


Authors

  • Maas, Jan
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 34E13 35Q84 37L45 60J28

Keywords

  • Reaction-rate equation, chemical master equation, Fokker-Planck equation, chemical Langevin dynamics, detailed-balance condition, relative entropy, dissipation potentials, gradient structures, many-particle limit

DOI

10.20347/WIAS.PREPRINT.2712

Abstract

We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.

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WIAS Preprint No. 2148, (2020)

Additive splitting methods for parallel solution of evolution problems


Authors

  • Amiranashvili, Shalva
    ORCID: 0000-0002-8132-882X
  • Radziunas, Mindaugas
    ORCID: 0000-0003-0306-1266
  • Bandelow, Uwe
    ORCID: 0000-0003-3677-2347
  • Busch, Kurt
  • Čiegis, Raimondas

2010 Mathematics Subject Classification

  • 41A25 65N12 65Y20 65Y05 68Q25 68W10

Keywords

  • Splitting method, Richardson extrapolation, nonlinear Schrödinger equation, nonlinear optics

DOI

10.20347/WIAS.PREPRINT.2767

Abstract

We demonstrate how a multiplicative splitting method of order P can be used to construct an additive splitting method of order P + 3. The weight coefficients of the additive method depend only on P, which must be an odd number. Specifically we discuss a fourth-order additive method, which is yielded by the Lie-Trotter splitting. We provide error estimates, stability analysis, and numerical examples with the special discussion of the parallelization properties and applications to nonlinear optics.

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WIAS Preprint No. 2148, (2020)

Multiband k $cdot$ p model and fitting scheme for ab initio-based electronic structure parameters for wurtzite GaAs


Authors

  • Marquardt, Oliver
    ORCID: 0000-0002-6642-8988
  • Caro, Miguel A.
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412
  • Mathé, Peter
    ORCID: 0000-0002-1208-1421
  • Willatzen, Morten

Keywords

  • Electronic bandstructure, quasi-Monte Carlo methods, k · p models, compound semiconductors

DOI

10.20347/WIAS.PREPRINT.2699

Abstract

We develop a 16-band k · p model for the description of wurtzite GaAs, together with a novel scheme to determine electronic structure parameters for multiband k · p models. Our approach uses low-discrepancy sequences to fit k · p band structures beyond the eight-band scheme to most recent ab initio data, obtained within the framework for hybrid-functional density functional theory with a screened-exchange hybrid functional. We report structural parameters, elastic constants, band structures along high-symmetry lines, and deformation potentials at the Γ point. Based on this, we compute the bulk electronic properties (Γ point energies, effective masses, Luttinger-like parameters, and optical matrix parameters) for a ten-band and a sixteen-band k · p model for wurtzite GaAs. Our fitting scheme can assign priorities to both selected bands and k points that are of particular interest for specific applications. Finally, ellipticity conditions can be taken into account within our fitting scheme in order to make the resulting parameter sets robust against spurious solutions.

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WIAS Preprint No. 2148, (2020)

Generalized iterated-sums signatures


Authors

  • Diehl, Joscha
  • Ebrahimi-Fard, Kurusch
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2010 Mathematics Subject Classification

  • 60L10 60L70 16Y60

Keywords

  • Time series analysis, time warping, signature, quasi-shuffle product, Hoffman`s exponential, Hopf algebra

DOI

10.20347/WIAS.PREPRINT.2795

Abstract

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.

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WIAS Preprint No. 2148, (2020)

The moving frame method for iterated-integrals: Orthogonal invariants


Authors

  • Diehl, Joscha
  • Preiß, Rosa
  • Ruddy, Michael
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2010 Mathematics Subject Classification

  • 60L10 14L24

Keywords

  • Signature, geometric invariants, moving frame, orthogonal group

DOI

10.20347/WIAS.PREPRINT.2796

Abstract

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Kiraly and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.

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WIAS Preprint No. 2148, (2020)

Optimal control and directional differentiability for elliptic quasi-variational inequalities


Authors

  • Alphonse, Amal
    ORCID: 0000-0001-7616-3293
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Rautenberg, Carlos N.
    ORCID: 0000-0001-9497-9296

2010 Mathematics Subject Classification

  • 47J20 49J40 49J52 49J50 49J21 49K21

Keywords

  • Quasi-variational inequality, obstacle problem, directional differentiability, optimal control, stationarity condition

DOI

10.20347/WIAS.PREPRINT.2756

Abstract

We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general unsigned data, thereby extending the results of our previous work which provided a first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.

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WIAS Preprint No. 2148, (2020)

Analysis of a quasi-variational contact problem arising in thermoelasticity


Authors

  • Alphonse, Amal
  • Rautenberg, Carlos N.
    ORCID: 0000-0001-9497-9296
  • Rodrigues, José Francisco
    ORCID: 0000-0001-8438-0749

2010 Mathematics Subject Classification

  • 35J47 35K40 35J87 35K86 35B65 80M30

Keywords

  • Elliptic-parabolic system, quasi-variational inequality, obstacle problem, thermoforming

DOI

10.20347/WIAS.PREPRINT.2747

Abstract

We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.

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WIAS Preprint No. 2148, (2020)

Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator


Authors

  • Heida, Martin
  • Kantner, Markus
    ORCID: 0000-0003-4576-3135
  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2010 Mathematics Subject Classification

  • 35Q84 49M25 65N08

Keywords

  • Finite volume, Fokker--Planck, Scharfetter--Gummel, Stolarsky mean, consistency, order of convergence

DOI

10.20347/WIAS.PREPRINT.2684

Abstract

We introduce a family of various finite volume discretization schemes for the Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter--Gummel scheme stands out compared to the others.

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WIAS Preprint No. 2148, (2020)

On primal and dual approaches for distributed stochastic convex optimization over networks


Authors

  • Dvinskikh, Darina
  • Gorbunov, Eduard
  • Gasnikov, Alexander
  • Dvurechensky, Alexander
  • Uribe, César A.

2010 Mathematics Subject Classification

  • 90C25 90C06 90C90

Keywords

  • Convex and non-convex optimization, stochastic optimization, first-order method, adaptive method, gradient descent, complexity bounds, mini-batch

DOI

10.20347/WIAS.PREPRINT.2690

Abstract

We introduce a primal-dual stochastic gradient oracle method for distributed convex optimization problems over networks. We show that the proposed method is optimal in terms of communication steps. Additionally, we propose a new analysis method for the rate of convergence in terms of duality gap and probability of large deviations. This analysis is based on a new technique that allows to bound the distance between the iteration sequence and the optimal point. By the proper choice of batch size, we can guarantee that this distance equals (up to a constant) to the distance between the starting point and the solution.

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WIAS Preprint No. 2148, (2020)

Malware propagation in urban D2D networks


Authors

  • Hinsen, Alexander
    ORCID: 0000-0001-6333-4962
  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Cali, Eli
  • Wary, Jean-Philippe

2010 Mathematics Subject Classification

  • 60J25 60K35 60K37

Keywords

  • Random environment, Cox--Gilbert graph, Poisson--Voronoi tessellation, interacting particle system, ad-hoc network, data propagation, white knight, speed of propagation, survival, extinction

DOI

10.20347/WIAS.PREPRINT.2674

Abstract

We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

Appeared in

  • IEEE 18th International Symposium on on Modeling and Optimization in Mobile, ad Hoc, and Wireless Networks, (WiOpt), Volos, Greece, Institute of Electrical and Electronics Engineers (IEEE), 2020, pp. 1--9.

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WIAS Preprint No. 2148, (2020)

Bifurcation structure of a swept source laser


Authors

  • Kovalev, Anton V.
  • Dmitriev, Pavel S.
  • Vladimirov, Andrei G.
  • Pimenov, Alexander
  • Huyet, Guillaume
  • Viktorov, Evgeniy A.

2010 Physics and Astronomy Classification Scheme

  • 42.55.Px, 42.65.Sf, 42.60.Mi, 42.55.Ah, 02.30.Ks

Keywords

  • Frequency swept souce, bifurcation structure

DOI

10.20347/WIAS.PREPRINT.2681

Abstract

We numerically analyze a delay differential equation model of a short-cavity semiconductor laser with an intracavity frequency swept filter and reveal a complex bifurcation structure responsible for the asymmetry of the output characteristics of this laser. We show that depending on the direction of the frequency sweep of a narrowband filter, there exist two bursting cycles determined by different parts of a continuous-wave solutions branch.

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WIAS Preprint No. 2148, (2020)

EDP-convergence for a linear reaction-diffusion system with fast reversible reaction


Authors

  • Stephan, Artur
    ORCID: 0000-0001-9871-3946

2010 Mathematics Subject Classification

  • 49S05 47J30 35A15 5K57 92E20 35Q84

Keywords

  • Markov process with detailed balance, linear reaction-diffusion system, gradient systems, gradient flows, evolutionary convergence, Energy-Dissipation-Balance, coarse-graining, microscopic equilibrium, Gamma-convergence

DOI

10.20347/WIAS.PREPRINT.2793

Abstract

We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.

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WIAS Preprint No. 2148, (2020)

Numerical simulation of TEM images for In(Ga)As/GaAs quantum dots with various shapes


Authors

  • Maltsi, Anieza
    ORCID: 0000-0003-2417-8770
  • Niermann, Tore
  • Streckenbach, Timo
    ORCID: 0009-0001-0874-0463
  • Tabelow, Karsten
    ORCID: 0000-0003-1274-9951
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412

2010 Mathematics Subject Classification

  • 34B60 74B10 78A45 81V65

Keywords

  • Semiconductors, quantum dots, TEM images, electron wave propagation, strain

DOI

10.20347/WIAS.PREPRINT.2682

Abstract

We present a mathematical model and a tool chain for the numerical simulation of TEM images of semiconductor quantum dots (QDs). This includes elasticity theory to obtain the strain profile coupled with the Darwin-Howie-Whelan equations, describing the propagation of the electron wave through the sample. We perform a simulation study on indium gallium arsenide QDs with different shapes and compare the resulting TEM images to experimental ones. This tool chain can be applied to generate a database of simulated TEM images, which is a key element of a novel concept for model-based geometry reconstruction of semiconductor QDs, involving machine learning techniques.

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WIAS Preprint No. 2148, (2020)

Two-phase flows for sedimentation of suspensions


Authors

  • Peschka, Dirk
    ORCID: 0000-0002-3047-1140
  • Rosenau, Matthias

2010 Mathematics Subject Classification

  • 35M33 76T20 76M10 76N99

Keywords

  • Two-phase suspension flow, free boundary problem, non-smooth dissipation, generalized gradient structure based on flow maps

DOI

10.20347/WIAS.PREPRINT.2743

Abstract

We present a two-phase flow model that arises from energetic-variational arguments and study its implication for the sedimentation of buoyant particles in a viscous fluid inside a Hele--Shaw cell and also compare corresponding simulation results to experiments. Based on a minimal dissipation argument, we provide a simplified 1D model applicable to sedimentation and study its properties and the numerical discretization. We also explore different aspects of its numerical discretization in 2D. The focus is on different possible stabilization techniques and their impact on the qualitative behavior of solutions. We use experimental data to verify some first qualitative model predictions and discuss these experiments for different stages of batch sedimentation.

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WIAS Preprint No. 2148, (2020)

Log-modulated rough stochastic volatility models


Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Harang, Fabian
  • Pigato, Paolo

2010 Mathematics Subject Classification

  • 91G30 60G22

Keywords

  • Rough volatility models, stochastic volatility, rough Bergomi model, implied skew, fractional Brownian motion, log Brownian motion

DOI

10.20347/WIAS.PREPRINT.2752

Abstract

We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index H. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for H = 0. As a consequence, the resulting super-rough stochastic volatility models can be analysed over the whole range of Hurst indices between 0 and 1/2, including H = 0, without the need of further normalization. We obtain the usual power law explosion of the skew as maturity T goes to 0, modulated by a logarithmic term, so no flattening of the skew occurs as H goes to 0.

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WIAS Preprint No. 2148, (2020)

Optimal stopping with signatures


Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Hager, Paul
  • Riedel, Sebastian
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266

2010 Mathematics Subject Classification

  • 60L10 60G40

Keywords

  • Signatures, rough paths, optimal stopping

DOI

10.20347/WIAS.PREPRINT.2790

Abstract

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process. We consider classic and randomized stopping times represented by linear functionals of the associated rough path signature, and prove that maximizing over the class of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. The only assumption on the process is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion which fail to be either semi-martingales or Markov processes.

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WIAS Preprint No. 2148, (2020)

Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity


Authors

  • Taggi, Lorenzo

2010 Mathematics Subject Classification

  • 82C22 60K35 82C26

Keywords

  • Essential enhancements, activated random walks, Abelian networks, self-organised criticality, absorbing-state phase transition

DOI

10.20347/WIAS.PREPRINT.2722

Abstract

We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of `increasing' events, extending previous results of Rolla and Sidoravicius (2012). Our proof method is of independent interest and can be viewed as a reformulation of the `essential enhancements' technique -- which was introduced for percolation -- in the framework of Abelian networks.

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WIAS Preprint No. 2148, (2020)

Exponential decay of transverse correlations for spin systems with continuous symmetry and non-zero external field


Authors

  • Lees, Benjamin
  • Taggi, Lorenzo

2010 Mathematics Subject Classification

  • 82B20 60K35 82B26

Keywords

  • Spin systems, phase transitions, statistical mechanics

DOI

10.20347/WIAS.PREPRINT.2730

Abstract

We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a `colour-switch' lemma, and a sampling procedure which allows us to bound from above the `typical' length of the open paths.

Appeared in

  • Probab. Theory Related Fields, 180 (2021), pp. 1099-1133.

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WIAS Preprint No. 2148, (2020)

Turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold


Authors

  • Gowda, Uday
  • Roche, Amy
  • Pimenov, Alexander
  • Vladimirov, Andrei G.
    ORCID: 0000-0002-7540-8380
  • Slepneva, Svetlana
  • Viktorov, Evgeny A.
  • Huyet, Guillaume

2010 Physics and Astronomy Classification Scheme

  • 42.55.Px 42.65.Sf 42.60.Mi 42.55.Ah 02.30.Ks

Keywords

  • Coherent structures, turbulence, long cavity laser

DOI

10.20347/WIAS.PREPRINT.2724

Abstract

We report on the formation of novel turbulent coherent structures in a long cavity semiconductor laser near the lasing threshold. Experimentally, the laser emits a series of power dropouts within a roundtrip and the number of dropouts per series depends on a set of parameters including the bias current. At fixed parameters, the drops remain dynamically stable, repeating over many roundtrips. By reconstructing the laser electric field in the case where the laser emits one dropout per round trip and simulating its dynamics using a time-delayed model, we discuss the reasons for long-term sustainability of these solutions. We suggest that the observed dropouts are closely related to the coherent structures of the cubic complex Ginzburg-Landau equation.

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WIAS Preprint No. 2148, (2020)

Runge--Kutta methods for rough differential equations


Authors

  • Redmann, Martin
    ORCID: 0000-0001-5182-9773
  • Riedel, Sebastian

2010 Mathematics Subject Classification

  • 60H10 60H35 65H30

Keywords

  • B-series, rough paths, Runge-Kutta methods

DOI

10.20347/WIAS.PREPRINT.2708

Abstract

We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (B-series) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying Runge-Kutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a Runge-Kutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivative-free. We provide a full characterization of this implementable Runge-Kutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index 1/4 < H ≤ 1/2. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence.

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WIAS Preprint No. 2148, (2020)

Semi-implicit Taylor schemes for stiff rough differential equations


Authors

  • Riedel, Sebastian

2010 Mathematics Subject Classification

  • 60G15 60H10 65C30

Keywords

  • Rough paths, semi-implicit Taylor schemes, stiff systems, stochastic differential equations

DOI

10.20347/WIAS.PREPRINT.2734

Abstract

We study a class of semi-implicit Taylor-type numerical methods that are easy to implement and designed to solve multidimensional stochastic differential equations driven by a general rough noise, e.g. a fractional Brownian motion. In the multiplicative noise case, the equation is understood as a rough differential equation in the sense of T. Lyons. We focus on equations for which the drift coefficient may be unbounded and satisfies a one-sided Lipschitz condition only. We prove well-posedness of the methods, provide a full analysis, and deduce their convergence rate. Numerical experiments show that our schemes are particularly useful in the case of stiff rough stochastic differential equations driven by a fractional Brownian motion.

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WIAS Preprint No. 2148, (2020)

Dynamics of a stochastic excitable system with slowly adapting feedback


Authors

  • Franović, Igor
  • Yanchuk, Serhiy
  • Eydam, Sebastian
  • Bačić, Iva
  • Wolfrum, Matthias
    ORCID: 0000-0002-4278-2675

2010 Physics and Astronomy Classification Scheme

  • 89.75.Fb, 05.40.Ca

Keywords

  • Excitability, adaption, coherence resonance

DOI

10.20347/WIAS.PREPRINT.2678

Abstract

We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic busting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance, or effectively control the features of the stochastic bursting. The setup can be considered as a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker-Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting.

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WIAS Preprint No. 2148, (2020)

Aging for the stationary Kardar--Parisi--Zhang equation and related models


Authors

  • Deuschel, Jean-Dominique
  • Orenshtein, Tal
  • Moreno Flores, Gregorio R.

2010 Mathematics Subject Classification

  • 60H15 35R60 82B44 82B26

Keywords

  • PZ equation, Cole-Hopf solution, time correlation, aging, space-time stationarity, directed polymers in random environment, last passage percolation, totally asymmetric exclusion process, Edwards-Wilkinson equation, Ginzburg-Landau model

DOI

10.20347/WIAS.PREPRINT.2763

Abstract

We study the aging property for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution to the stationary KPZ equation, the height function of the stationary TASEP, last-passage percolation with boundary conditions and stationary directed polymers in the intermediate disorder regime. All of these models are shown to display a universal aging behavior characterized by the rate of decay of their correlations. As a comparison, we show aging for models in the Edwards-Wilkinson universality class where a different decay exponent is obtained. A key ingredient to our proofs is a characteristic of space-time stationarity - covariance-to-variance reduction - which allows to deduce the asymptotic behavior of the correlations of two space-time points by the one of the variances at one point. We formulate several open problems.

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WIAS Preprint No. 2148, (2020)

Spatial decay of the vorticity field of time-periodic viscous flow past a body


Authors

  • Eiter, Thomas
    ORCID: 0000-0002-7807-1349
  • Galdi, Giovanni P.
    ORCID: 0000-0003-3812-9400

2010 Mathematics Subject Classification

  • 35Q30 35B10 76D05 35E05

Keywords

  • Navier-Stokes, time-periodic solutions, vorticity field, fundamental solution, asymptotic behavior

DOI

10.20347/WIAS.PREPRINT.2791

Abstract

We study the asymptotic spatial behavior of the vorticity field associated to a time-periodic Navier-Stokes flow past a body in the class of weak solutions satisfying a Serrin-like condition. We show that outside the wake region the vorticity field decays pointwise at an exponential rate, uniformly in time. Moreover, decomposing it into its time-average over a period and a so-called purely periodic part, we prove that inside the wake region, the time-average has the same algebraic decay as that known for the associated steady-state problem, whereas the purely periodic part decays even faster, uniformly in time. This implies, in particular, that ``sufficiently far'' from the body, the time-periodic vorticity field behaves like the vorticity field of the corresponding steady-state problem.

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WIAS Preprint No. 2148, (2020)

Fast reaction limits via $Gamma$-convergence of the flux rate functional


Authors

  • Peletier, Mark A.
  • Renger, D. R. Michiel
    ORCID: 0000-0003-3557-3485

2010 Mathematics Subject Classification

  • 05C21 34E05 35A15 60F10 60J27

Keywords

  • Fast reaction limit, quasi steady state approximation, Gamma convergence, finite graph

DOI

10.20347/WIAS.PREPRINT.2766

Abstract

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as 1/∈, and we prove the convergence in the fast-reaction limit ∈ → 0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

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WIAS Preprint No. 2148, (2020)

Stochastic homogenization on randomly perforated domains


Authors

  • Heida, Martin
    ORCID: 0000-0002-7242-8175

2010 Mathematics Subject Classification

  • 80M40 60D05

Keywords

  • Stochastic homogenization, stochastic geometry, p-Laplace

DOI

10.20347/WIAS.PREPRINT.2742

Abstract

We study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions on the geometry which we call local (δ, M)-regularity, isotropic cone mixing and bounded average connectivity. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. The third is the most tricky part and measures the ''mesoscopic'' connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process. We finally introduce suitable Sobolev spaces on Rd and Ω in order to construct a stochastic two-scale convergence method and apply the resulting theory to the homogenization of a p-Laplace problem on a randomly perforated domain.

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WIAS Preprint No. 2148, (2020)

The parabolic Anderson model on a Galton--Watson tree


Authors

  • den Hollander, Frank
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Soares dos Santos, Renato

2010 Mathematics Subject Classification

  • 05C80 60H25 82B44

Keywords

  • Parabolic Anderson model, random graphs, tree-like graphs, Galton--Watson tree, random walk in random potential, large-time asymptotics, almost-sure asymptotics, eigenvalues of random operators

DOI

10.20347/WIAS.PREPRINT.2675

Abstract

We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.

Appeared in

  • In and out of euilibrium 3: Celebrating Vladas Sidovaricius, M.E. Vares, R. Fernandez, L.R. Fontes, C.M. Newman, eds., vol. 77 of Progress in Probability, Birkhäuser, 2021, pp. X, 590, DOI 10.1007/978-3-030-60754-8 .

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WIAS Preprint No. 2148, (2020)

Distributed optimization with quantization for computing Wasserstein barycenters


Authors

  • Krawchenko, Roman
  • Uribe, César A.
  • Gasnikov, Alexander
  • Dvurechensky, Pavel
    ORCID: 0000-0003-1201-2343

2010 Mathematics Subject Classification

  • 90C25 90C30 90C06

Keywords

  • Distributed convex optimization, quantization, optimal transport, Wasserstein distance

DOI

10.20347/WIAS.PREPRINT.2782

Abstract

We study the problem of the decentralized computation of entropy-regularized semi-discrete Wasserstein barycenters over a network. Building upon recent primal-dual approaches, we propose a sampling gradient quantization scheme that allows efficient communication and computation of approximate barycenters where the factor distributions are stored distributedly on arbitrary networks. The communication and algorithmic complexity of the proposed algorithm are shown, with explicit dependency on the size of the support, the number of distributions, and the desired accuracy. Numerical results validate our algorithmic analysis.

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WIAS Preprint No. 2148, (2020)

Dynamical phase transitions for flows on finite graphs


Authors

  • Gabrielli, Davide
  • Renger, D. R. Michiel
    ORCID: 0000-0003-3557-3485

2010 Mathematics Subject Classification

  • 60F10 05C21 82C22 82C26

Keywords

  • Large deviations, particle systems, phase transitions

DOI

10.20347/WIAS.PREPRINT.2746

Abstract

We study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.

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WIAS Preprint No. 2148, (2020)

Iterated-sums signature, quasi-symmetric functions and time series analysis


Authors

  • Diehl, Joscha
  • Ebrahimi-Fard, Kurusch
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2010 Mathematics Subject Classification

  • 60L10 60L70

Keywords

  • Shuffle and quasi-shuffle Hopf algebras, half-shuffles, iterated integral- and sum-signatures, quasi-symmetric functions, Hoffmans's exponential

DOI

10.20347/WIAS.PREPRINT.2736

Abstract

We survey and extend results on a recently defined character on the quasi-shuffle algebra. This character, termed iterated-sums signature, appears in the context of time series analysis and originates from a problem in dynamic time warping. Algebraically, it relates to (multidimensional) quasisymmetric functions as well as (deformed) quasi-shuffle algebras.

Appeared in

  • Sem. Lothar. Combin., 84B (2020), pp. 86/1--86/12.

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WIAS Preprint No. 2148, (2020)

Micro-geometry effects on the nonlinear effective yield strength response of magnetorheological fluids


Authors

  • Nika, Grigor
    ORCID: 0000-0002-4403-6908
  • Vernescu, Bogdan
    ORCID: 0000-0001-6153-6392

2010 Mathematics Subject Classification

  • 35J57 35J60 35M12 76T20

Keywords

  • Magnetorheological fluids, chain structures, surface-to-volume effects

DOI

10.20347/WIAS.PREPRINT.2673

Abstract

We use the novel constitutive model in [15], derived using the homogenization method, to investigate the effect particle chain microstructures have on the properties of the magnetorheological fluid. The model allows to compute the constitutive coefficients for different geometries. Different geometrical realizations of chains can significantly change the magnetorheological effect of the suspension. Numerical simulations suggest that particle size is also important as the increase of the overall particle surface area can lead to a decrease of the overall magnetorheological effect while keeping the volume fraction constant.

Appeared in

  • Emerging Problems in the Homogenization of Partial Differential Equations, P. Donato, M. Luna-Laynez, eds., vol. 10 of SEMA SIMAI Springer Series, Springer, Cham, 2021, pp. 1--16, DOI 978-3-030-62030-1_1 .

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WIAS Preprint No. 2148, (2020)

Beyond just ``flattening the curve'': Optimal control of epidemics with purely non-pharmaceutical interventions


Authors

  • Kantner, Markus
    ORCID: 0000-0003-4576-3135
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412

2010 Mathematics Subject Classification

  • 92D30 37N25 37N40 93C10 49N90 34B15

Keywords

  • Mathematical epidemiology, optimal control, non-pharmaceutical interventions, effective reproduction number, dynamical systems, COVID-19, SARS-CoV2

DOI

10.20347/WIAS.PREPRINT.2748

Abstract

When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

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WIAS Preprint No. 2148, (2020)

Wick polynomials in non-commutative probability: A group-theoretical approach


Authors

  • Ebrahimi-Fard, Kurusch
  • Patras, Frédéric
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492
  • Zambotti, Lorenzo

2010 Mathematics Subject Classification

  • 16T05 16T10 16T30

Keywords

  • Wick polynomials, monotone cumulants, free cumulants, boolean cumulants, formal power series, combinatorial Hopf algebra, shuffle algebra, group actions

DOI

10.20347/WIAS.PREPRINT.2677

Abstract

Wick polynomials and Wick products are studied in the context of non-commutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf algebraic approach to cumulants and Wick products in classical probability theory.

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